[ 0s] Memory limit set to 100366504KB [ 0s] Using BUILD_ROOT=/data/workspace/gbsbuild-ROOT/GBS-ROOT/local/BUILD-ROOTS/scratch.i586.5 [ 0s] Using BUILD_ARCH=i686:i586:i486:i386:noarch [ 0s] [ 0s] [ 0s] ip-192-168-56-24 started "build lwnode.spec" at Wed Apr 24 06:22:50 UTC 2024. [ 0s] [ 0s] [ 0s] processing recipe /data/workspace/gbsbuild-ROOT/GBS-ROOT/local/sources/tizen_6.5_unified_emulator/lwnode-1.0.0-1/lwnode.spec ... [ 0s] init_buildsystem --configdir /usr/lib/build/configs --cachedir /data/workspace/gbsbuild-ROOT/GBS-ROOT/local/cache --define jobs 8 --define _smp_mflags -j8 --define _costomized_smp_mflags -j20 --repository /data/workspace/gbsbuild-ROOT/GBS-ROOT/local/repos/tizen_6.5_unified_emulator/i586/RPMS --repository http://download.tizen.org/snapshots/TIZEN/Tizen-6.5/Tizen-6.5-Base/tizen-6.5-base_20240329.041322/repos/standard/packages/ --repository http://download.tizen.org/snapshots/TIZEN/Tizen-6.5/Tizen-6.5-Base/tizen-6.5-base_20240329.041322/repos/standard/debug/ /data/workspace/gbsbuild-ROOT/GBS-ROOT/local/sources/tizen_6.5_unified_emulator/lwnode-1.0.0-1/lwnode.spec build ... [ 0s] initializing /data/workspace/gbsbuild-ROOT/GBS-ROOT/local/BUILD-ROOTS/scratch.i586.5/.srcfiles.cache ... [ 0s] /usr/lib/build/createdirdeps --oldfile /data/workspace/gbsbuild-ROOT/GBS-ROOT/local/BUILD-ROOTS/scratch.i586.5/.srcfiles.cache /data/workspace/gbsbuild-ROOT/GBS-ROOT/local/repos/tizen_6.5_unified_emulator/i586/RPMS [ 1s] /usr/lib/build/createrepomddeps --cachedir=/data/workspace/gbsbuild-ROOT/GBS-ROOT/local/cache http://download.tizen.org/snapshots/TIZEN/Tizen-6.5/Tizen-6.5-Base/tizen-6.5-base_20240329.041322/repos/standard/packages/ [ 2s] /usr/lib/build/createrepomddeps --cachedir=/data/workspace/gbsbuild-ROOT/GBS-ROOT/local/cache http://download.tizen.org/snapshots/TIZEN/Tizen-6.5/Tizen-6.5-Base/tizen-6.5-base_20240329.041322/repos/standard/debug/ [ 3s] expanding package dependencies... [ 3s] copying qemu [ 3s] [ 4s] calculating packages to download... [ 20s] [1/1] downloading http://download.tizen.org/snapshots/TIZEN/Tizen-6.5/Tizen-6.5-Base/tizen-6.5-base_20240329.041322/repos/standard/packages/i686/nghttp2-devel-1.41.0-1.1.i686.rpm ... [ 21s] [ 21s] reordering...cycle: gio-branding-upstream -> libgio [ 21s] breaking dependency libgio -> gio-branding-upstream [ 21s] cycle: pkgmgr-info -> pkgmgr-info-parser [ 21s] breaking dependency pkgmgr-info -> pkgmgr-info-parser [ 21s] cycle: libalarm -> alarm-server [ 21s] breaking dependency alarm-server -> libalarm [ 21s] cycle: libkey-manager-client -> key-manager [ 21s] breaking dependency key-manager -> libkey-manager-client [ 21s] cycle: libsecurity-manager-client -> security-manager [ 21s] breaking dependency security-manager -> libsecurity-manager-client [ 21s] cycle: libsecurity-manager-client -> security-manager -> security-license-manager [ 21s] breaking dependency security-manager -> security-license-manager [ 21s] cycle: security-manager -> security-manager-policy [ 21s] breaking dependency security-manager -> security-manager-policy [ 21s] cycle: xmlsec1-openssl -> xmlsec1 [ 21s] breaking dependency xmlsec1-openssl -> xmlsec1 [ 21s] cycle: libsoftokn3 -> nss [ 21s] breaking dependency nss -> libsoftokn3 [ 22s] reordering done [ 31s] deleting app2sd [ 32s] deleting app2sd-devel [ 32s] deleting boost-atomic [ 32s] deleting boost-chrono [ 33s] deleting boost-container [ 33s] deleting boost-context [ 33s] deleting boost-contract [ 34s] deleting boost-coroutine [ 34s] deleting boost-date-time [ 35s] deleting boost-devel [ 36s] deleting boost-fiber [ 37s] deleting boost-graph [ 37s] deleting boost-iostreams [ 38s] deleting boost-locale [ 38s] deleting boost-log [ 38s] deleting boost-math [ 39s] deleting boost-python [ 39s] deleting boost-random [ 40s] deleting boost-serialization [ 40s] deleting boost-test [ 40s] deleting boost-timer [ 41s] deleting boost-wave [ 41s] deleting boost_stacktrace [ 41s] deleting boost_type_erasure [ 42s] deleting cert-svc-devel [ 42s] deleting cryptsetup [ 42s] deleting dbus [ 42s] deleting dbus-libs [ 42s] deleting device-mapper [ 43s] deleting gtest [ 43s] /sbin/ldconfig: relative path `0' used to build cache [ 43s] warning: %postun(gtest-1.10.0-0.i686) scriptlet failed, exit status 1 [ 43s] deleting gtest-devel [ 43s] deleting gumd [ 44s] deleting klay [ 44s] /sbin/ldconfig: relative path `0' used to build cache [ 44s] warning: %postun(klay-0.0.2-0.i686) scriptlet failed, exit status 1 [ 44s] deleting libdevmapper [ 44s] deleting libgcrypt-devel [ 44s] deleting libgpg-error-devel [ 44s] deleting libgum [ 45s] deleting libgum-devel [ 45s] deleting libiri [ 45s] deleting libiri-devel [ 45s] deleting libsecurity-manager-client-devel [ 45s] deleting libstorage-devel [ 45s] deleting libtzplatform-config-devel [ 45s] deleting libxml2-devel [ 45s] deleting libxml2-tools [ 45s] deleting libxslt-devel [ 45s] deleting libxslt-tools [ 45s] deleting libzip [ 45s] deleting libzip-devel [ 45s] deleting manifest-parser [ 46s] deleting manifest-parser-devel [ 46s] deleting minizip-devel [ 46s] deleting ncurses-devel [ 46s] deleting pkgmgr-client-devel [ 46s] deleting pkgmgr-info-devel [ 46s] deleting pkgmgr-info-parser-devel [ 46s] deleting pkgmgr-installer-devel [ 46s] deleting pkgmgr-types-devel [ 46s] deleting readline-devel [ 46s] deleting security-privilege-manager-devel [ 46s] deleting smack-devel [ 46s] deleting systemd-devel [ 46s] deleting tpk-manifest-handlers [ 47s] deleting tpk-manifest-handlers-devel [ 47s] deleting trust-anchor [ 47s] /sbin/ldconfig: relative path `0' used to build cache [ 47s] warning: %postun(trust-anchor-2.1.2-0.i686) scriptlet failed, exit status 1 [ 47s] deleting trust-anchor-devel [ 47s] deleting ttrace-devel [ 47s] deleting xmlsec1-devel [ 47s] deleting xmlsec1-gcrypt [ 47s] deleting xz-devel [ 48s] [1/235] keeping boost-license-1.71.0-1.1 [ 48s] [2/235] keeping ca-certificates-tizen-0.2.0-0 [ 48s] [3/235] keeping default-ac-domains-0.1-1.1 [ 48s] [4/235] keeping dlog-config-persistent-logs-6.5.11-1 [ 48s] [5/235] keeping libmagic-data-5.39-1.9 [ 48s] [6/235] keeping setup-0.9-1.2 [ 48s] [7/235] keeping filesystem-3.1-1.1 [ 48s] [8/235] keeping 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capi-appfw-app-common-0.8.11-0 [ 51s] [225/235] cumulate capi-appfw-app-control-0.15.4-0 [ 51s] [226/235] keeping security-license-manager-1.7.19-0 [ 51s] [227/235] keeping key-manager-0.1.58.5-1 [ 51s] [228/235] cumulate capi-appfw-app-common-devel-0.8.11-0 [ 51s] [229/235] keeping libkey-manager-client-0.1.58.5-1 [ 51s] [230/235] cumulate notification-0.5.95-1 [ 51s] [231/235] keeping cert-svc-2.2.1-0 [ 51s] [232/235] cumulate alarm-server-0.12.15-1 [ 51s] [233/235] cumulate libalarm-0.12.15-1 [ 51s] [234/235] cumulate capi-system-system-settings-0.0.5-0 [ 51s] [235/235] cumulate capi-system-system-settings-devel-0.0.5-0 [ 51s] now installing cumulated packages [ 52s] Preparing... ######################################## [ 52s] Updating / installing... [ 52s] libavutil-4.3.1-4 ######################################## [ 53s] libavcodec-4.3.1-4 ######################################## [ 53s] app-svc-0.1.57-0 ######################################## [ 53s] capi-appfw-package-manager-0.10.3-1 ######################################## [ 53s] libavformat-4.3.1-4 ######################################## [ 54s] libswscale-4.3.1-4 ######################################## [ 54s] capi-appfw-app-manager-0.12.2-1 ######################################## [ 54s] eventsystem-0.1.15-1 ######################################## [ 54s] hal-api-common-0.0.1-1 ######################################## [ 54s] hal-api-device-1.0.0-0 ######################################## [ 54s] libcynara-creds-gdbus-0.18.1-1 ######################################## [ 55s] libmm-common-0.2.128-0 ######################################## [ 55s] libmm-fileinfo-1.0.7-0 ######################################## [ 55s] capi-media-metadata-extractor-0.1.17-1######################################## [ 55s] libtracker-1.0.1-1 ######################################## [ 55s] capi-system-device-0.1.0-0 ######################################## [ 56s] nghttp2-1.41.0-1.1 ######################################## [ 56s] db4-4.8.30.NC-1.10 ######################################## [ 56s] sensord-dummy-4.0.54-1 ######################################## [ 56s] app-core-common-1.15.4-0 ######################################## [ 56s] capi-appfw-app-common-0.8.11-0 ######################################## [ 57s] capi-appfw-app-control-0.15.4-0 ######################################## [ 57s] notification-0.5.95-1 ######################################## [ 57s] alarm-server-0.12.15-1 ######################################## [ 57s] libalarm-0.12.15-1 ######################################## [ 57s] capi-system-system-settings-0.0.5-0 ######################################## [ 58s] libpython-2.7.18-1.10 ######################################## [ 58s] libcares-1.17.1-1.1 ######################################## [ 58s] libcares-devel-1.17.1-1.1 ######################################## [ 58s] capi-system-system-settings-devel-0.0.######################################## [ 58s] capi-appfw-app-common-devel-0.8.11-0 ######################################## [ 58s] nghttp2-devel-1.41.0-1.1 ######################################## [ 58s] libicu-devel-67.1-1.3 ######################################## [ 60s] python-2.7.18-1.10 ######################################## [ 60s] ninja-1.10.2-1.1 ######################################## [ 60s] zip-3.0-1.1 ######################################## [ 60s] Running alarm-server %posttrans script [ 60s] removing nis flags from /data/workspace/gbsbuild-ROOT/GBS-ROOT/local/BUILD-ROOTS/scratch.i586.5/etc/nsswitch.conf... [ 60s] now finalizing build dir... [ 61s] ----------------------------------------------------------------- [ 61s] ----- building lwnode.spec (user abuild) [ 61s] ----------------------------------------------------------------- [ 61s] ----------------------------------------------------------------- [ 62s] + exec rpmbuild --define 'jobs 8' --define '_smp_mflags -j8' --define '_costomized_smp_mflags -j20' --define '_srcdefattr (-,root,root)' --nosignature --target=i686-tizen-linux --define '_build_create_debug 1' -ba /home/abuild/rpmbuild/SOURCES/lwnode.spec [ 62s] Building target platforms: i686-tizen-linux [ 62s] Building for target i686-tizen-linux [ 62s] Executing(%prep): /bin/sh -e /var/tmp/rpm-tmp.Etuq0z [ 62s] + umask 022 [ 62s] + cd /home/abuild/rpmbuild/BUILD [ 62s] + cd /home/abuild/rpmbuild/BUILD [ 62s] + rm -rf lwnode-1.0.0 [ 62s] + /bin/tar -xof - [ 62s] + /bin/gzip -dc /home/abuild/rpmbuild/SOURCES/lwnode-1.0.0.tar.gz [ 64s] + STATUS=0 [ 64s] + '[' 0 -ne 0 ']' [ 64s] + cd lwnode-1.0.0 [ 64s] + /bin/chmod -Rf a+rX,u+w,g-w,o-w . [ 64s] + exit 0 [ 64s] Executing(%build): /bin/sh -e /var/tmp/rpm-tmp.ImKhWA [ 64s] + umask 022 [ 64s] + cd /home/abuild/rpmbuild/BUILD [ 64s] + cd lwnode-1.0.0 [ 64s] ++ uname -m [ 64s] + [[ i686 == \a\a\r\c\h\6\4 ]] [ 64s] + LANG=C [ 64s] + export LANG [ 64s] + unset DISPLAY [ 64s] + CFLAGS='-O2 -g2 -pipe -Wall -Wp,-D_FORTIFY_SOURCE=2 -fexceptions -fstack-protector-strong -Wformat-security -fmessage-length=0 -frecord-gcc-switches -Wl,-z,relro,--as-needed -feliminate-unused-debug-types --param=ssp-buffer-size=4 -fdiagnostics-color=never -m32 -march=i686 -mtune=i686 -msse4.2 -mfpmath=sse -fasynchronous-unwind-tables -fno-omit-frame-pointer -mstackrealign -Wl,-z,noseparate-code -g' [ 64s] + export CFLAGS [ 64s] + CXXFLAGS='-O2 -g2 -pipe -Wall -Wp,-D_FORTIFY_SOURCE=2 -fexceptions -fstack-protector-strong -Wformat-security -fmessage-length=0 -frecord-gcc-switches -Wl,-z,relro,--as-needed -feliminate-unused-debug-types --param=ssp-buffer-size=4 -fdiagnostics-color=never -m32 -march=i686 -mtune=i686 -msse4.2 -mfpmath=sse -fasynchronous-unwind-tables -fno-omit-frame-pointer -mstackrealign -Wl,-z,noseparate-code -g' [ 64s] + export CXXFLAGS [ 64s] + FFLAGS='-O2 -g2 -pipe -Wall -Wp,-D_FORTIFY_SOURCE=2 -fexceptions -fstack-protector-strong -Wformat-security -fmessage-length=0 -frecord-gcc-switches -Wl,-z,relro,--as-needed -feliminate-unused-debug-types --param=ssp-buffer-size=4 -fdiagnostics-color=never -m32 -march=i686 -mtune=i686 -msse4.2 -mfpmath=sse -fasynchronous-unwind-tables -fno-omit-frame-pointer -mstackrealign -Wl,-z,noseparate-code -g -I%_fmoddir' [ 64s] + export FFLAGS [ 64s] + LDFLAGS= [ 64s] + export LDFLAGS [ 64s] + gcc --version [ 64s] gcc (Tizen GCC 9.2.0 20190812 1.11) 9.2.0 [ 64s] Copyright (C) 2019 Free Software Foundation, Inc. [ 64s] This is free software; see the source for copying conditions. There is NO [ 64s] warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. [ 64s] [ 64s] + echo -e '\033[0;32mproduction\033[0m' [ 64s] production [ 64s] + echo Building: lwnode [ 64s] Building: lwnode [ 64s] + CFLAGS+=' -Os ' [ 64s] + CXXFLAGS+=' -Os ' [ 64s] + ./configure --tizen --without-npm --without-inspector --without-node-code-cache --without-node-snapshot --with-intl none --shared-zlib --shared-cares --shared-openssl --shared-nghttp2 --enable-reload-script --enable-external-builtin-script --dest-os linux --dest-cpu x32 --ninja --without-bundled-v8 --engine escargot --shared [ 65s] Package openssl was not found in the pkg-config search path. [ 65s] Perhaps you should add the directory containing `openssl.pc' [ 65s] to the PKG_CONFIG_PATH environment variable [ 65s] No package 'openssl' found [ 65s] Node.js configure: Found Python 3.9.1... [ 65s] WARNING: No acceptable ld.gold linker found! [ 65s] LWNode.js options: [-Dbuild_mode=release -Dlwnode=true -Dlwnode_jsengine_path=lwnode/code/escargotshim -Denable_external_builtin_script=true -Denable_reload_script=true -Dtarget_os=tizen -Dprofile=common -Descargot_lib_type=shared_lib -Descargot_threading=0 -Dnode_core_target_name=lwnode -Dnode_lib_target_name=liblwnode] [ 65s] WARNING: warnings were emitted in the configure phase [ 65s] INFO: configure completed successfully [ 65s] + ninja -C out/tizen/Release liblwnode [ 65s] ninja: Entering directory `out/tizen/Release' [ 65s] [1/231] CC obj/deps/uvwasi/src/uvwasi.wasi_rights.o [ 65s] [2/231] CC obj/deps/llhttp/src/llhttp.http.o [ 66s] [3/231] CC obj/deps/brotli/c/common/brotli.constants.o [ 66s] [4/231] CC obj/deps/uv/src/unix/libuv.getnameinfo.o [ 66s] [5/231] CC obj/deps/uv/src/unix/libuv.random-devurandom.o [ 66s] [6/231] CC obj/deps/uvwasi/src/uvwasi.wasi_serdes.o [ 66s] [7/231] CC obj/deps/uv/src/unix/libuv.getaddrinfo.o [ 66s] [8/231] CC obj/deps/uv/src/unix/libuv.loop.o [ 66s] [9/231] CC obj/deps/uv/src/unix/libuv.poll.o [ 66s] [10/231] CC obj/deps/uv/src/unix/libuv.procfs-exepath.o [ 66s] [11/231] CC obj/deps/uv/src/unix/libuv.random-getrandom.o [ 66s] [12/231] CC obj/deps/uvwasi/src/uvwasi.clocks.o [ 66s] [13/231] CC obj/deps/brotli/c/common/brotli.context.o [ 66s] [14/231] CC obj/deps/brotli/c/common/brotli.platform.o [ 66s] [15/231] CC obj/deps/histogram/src/histogram.hdr_histogram.o [ 66s] [16/231] CC obj/deps/uv/src/unix/libuv.loop-watcher.o [ 66s] [17/231] CC obj/deps/uv/src/unix/libuv.pipe.o [ 66s] [18/231] CC obj/deps/uv/src/unix/libuv.tcp.o [ 66s] [19/231] CC obj/deps/uv/src/unix/libuv.thread.o [ 66s] [20/231] CC obj/deps/uv/src/unix/libuv.tty.o [ 66s] [21/231] CC obj/deps/uvwasi/src/uvwasi.fd_table.o [ 66s] [22/231] CC obj/deps/uv/src/unix/libuv.random-sysctl-linux.o [ 66s] [23/231] CC obj/deps/uvwasi/src/uvwasi.uv_mapping.o [ 66s] [24/231] CC obj/deps/uvwasi/src/uvwasi.poll_oneoff.o [ 66s] [25/231] CC obj/deps/brotli/c/common/brotli.transform.o [ 66s] [26/231] CC obj/deps/brotli/c/dec/brotli.bit_reader.o [ 66s] [27/231] CC obj/deps/llhttp/src/llhttp.api.o [ 66s] [28/231] CC obj/deps/uv/src/libuv.strscpy.o [ 66s] [29/231] CC obj/deps/uv/src/unix/libuv.process.o [ 66s] [30/231] CC obj/deps/uv/src/unix/libuv.signal.o [ 66s] [31/231] CC obj/deps/uvwasi/src/uvwasi.path_resolver.o [ 66s] [32/231] CC obj/deps/uv/src/libuv.fs-poll.o [ 66s] [33/231] CC obj/deps/uv/src/libuv.inet.o [ 66s] [34/231] CC obj/deps/uv/src/libuv.random.o [ 66s] [35/231] CC obj/deps/uv/src/libuv.threadpool.o [ 66s] [36/231] CC obj/deps/uv/src/libuv.timer.o [ 66s] [37/231] CC obj/deps/uv/src/libuv.uv-data-getter-setters.o [ 66s] [38/231] CC obj/deps/uv/src/libuv.version.o [ 66s] [39/231] CC obj/deps/uv/src/unix/libuv.async.o [ 66s] [40/231] AR obj/deps/histogram/libhistogram.a [ 66s] [41/231] CC obj/deps/uv/src/unix/libuv.dl.o [ 66s] [42/231] CC obj/lwnode/code/escargotshim/deps/minizip/minizip.ioapi.o [ 66s] [43/231] CC obj/deps/uv/src/unix/libuv.linux-syscalls.o [ 66s] [44/231] CC obj/deps/uvwasi/src/uvwasi.uvwasi.o [ 66s] [45/231] CC obj/deps/uv/src/unix/libuv.stream.o [ 66s] [46/231] CC obj/deps/uv/src/unix/libuv.udp.o [ 66s] [47/231] CC obj/deps/brotli/c/common/brotli.dictionary.o [ 66s] [48/231] CC obj/deps/uv/src/libuv.idna.o [ 66s] [49/231] CC obj/deps/uv/src/unix/libuv.proctitle.o [ 66s] [50/231] CC obj/deps/brotli/c/dec/brotli.huffman.o [ 66s] [51/231] CC obj/deps/brotli/c/dec/brotli.state.o [ 66s] [52/231] CC obj/deps/uv/src/unix/libuv.linux-inotify.o [ 66s] [53/231] CC obj/deps/brotli/c/enc/brotli.fast_log.o [ 66s] [54/231] AR obj/deps/uvwasi/libuvwasi.a [ 66s] [55/231] CC obj/deps/brotli/c/enc/brotli.memory.o [ 66s] [56/231] CC obj/deps/uv/src/unix/libuv.linux-core.o [ 66s] [57/231] CC obj/deps/brotli/c/enc/brotli.utf8_util.o [ 66s] [58/231] CC obj/deps/brotli/c/enc/brotli.entropy_encode.o [ 67s] [59/231] CC obj/deps/brotli/c/enc/brotli.literal_cost.o [ 67s] [60/231] CC obj/deps/uv/src/libuv.uv-common.o [ 67s] [61/231] CC obj/lwnode/code/escargotshim/deps/minizip/minizip.unzip.o [ 67s] [62/231] CC obj/deps/brotli/c/enc/brotli.dictionary_hash.o [ 67s] [63/231] AR obj/lwnode/code/escargotshim/deps/minizip/libminizip.a [ 67s] [64/231] CC obj/deps/uv/src/unix/libuv.fs.o [ 67s] ../../../deps/uv/src/unix/fs.c: In function 'uv__fs_sendfile': [ 67s] ../../../deps/uv/src/unix/fs.c:909:57: warning: passing argument 4 of 'uv__fs_copy_file_range' from incompatible pointer type [-Wincompatible-pointer-types] [ 67s] 909 | r = uv__fs_copy_file_range(in_fd, NULL, out_fd, &off, req->bufsml[0].len, 0); [ 67s] | ^~~~ [ 67s] | | [ 67s] | off_t * {aka long long int *} [ 67s] In file included from ../../../deps/uv/src/unix/internal.h:41, [ 67s] from ../../../deps/uv/src/unix/fs.c:30: [ 67s] ../../../deps/uv/src/unix/linux-syscalls.h:71:33: note: expected 'ssize_t *' {aka 'int *'} but argument is of type 'off_t *' {aka 'long long int *'} [ 67s] 71 | ssize_t* off_out, [ 67s] | ~~~~~~~~~^~~~~~~ [ 67s] ../../../deps/uv/src/unix/fs.c: In function 'uv__fs_copyfile': [ 67s] ../../../deps/uv/src/unix/fs.c:1223:20: warning: comparison of integer expressions of different signedness: '__fsword_t' {aka 'int'} and 'unsigned int' [-Wsign-compare] [ 67s] 1223 | if (s.f_type != /* CIFS */ 0xFF534D42u) [ 67s] | ^~ [ 67s] [65/231] CC obj/deps/brotli/c/enc/brotli.command.o [ 67s] [66/231] CC obj/deps/uv/src/unix/libuv.core.o [ 67s] [67/231] CC obj/deps/llhttp/src/llhttp.llhttp.o [ 67s] ../../../deps/llhttp/src/llhttp.c: In function 'llhttp__internal__run': [ 67s] ../../../deps/llhttp/src/llhttp.c:8691:13: warning: unused variable 'avail' [-Wunused-variable] [ 67s] 8691 | int avail; [ 67s] | ^~~~~ [ 67s] ../../../deps/llhttp/src/llhttp.c:8869:13: warning: unused variable 'avail' [-Wunused-variable] [ 67s] 8869 | int avail; [ 67s] | ^~~~~ [ 67s] ../../../deps/llhttp/src/llhttp.c:9619:13: warning: unused variable 'avail' [-Wunused-variable] [ 67s] 9619 | int avail; [ 67s] | ^~~~~ [ 67s] [68/231] AR obj/deps/uv/libuv.a [ 67s] [69/231] CC obj/deps/brotli/c/enc/brotli.encoder_dict.o [ 67s] [70/231] AR obj/deps/llhttp/libllhttp.a [ 67s] [71/231] CC obj/deps/brotli/c/enc/brotli.histogram.o [ 67s] [72/231] CC obj/deps/brotli/c/enc/brotli.bit_cost.o [ 68s] [73/231] CC obj/deps/brotli/c/enc/brotli.static_dict.o [ 68s] [74/231] ACTION escargot: config escargot_77aa8dffe56dc6e9e1faabd74ff0cddd [ 68s] -- The C compiler identification is GNU 9.2.0 [ 68s] -- The CXX compiler identification is GNU 9.2.0 [ 68s] -- Check for working C compiler: /bin/cc [ 68s] -- Check for working C compiler: /bin/cc -- works [ 68s] -- Detecting C compiler ABI info [ 68s] -- Detecting C compiler ABI info - done [ 68s] -- Detecting C compile features [ 68s] -- Detecting C compile features - done [ 68s] -- Check for working CXX compiler: /bin/c++ [ 68s] -- Check for working CXX compiler: /bin/c++ -- works [ 68s] -- Detecting CXX compiler ABI info [ 68s] -- Detecting CXX compiler ABI info - done [ 68s] -- Detecting CXX compile features [ 68s] -- Detecting CXX compile features - done [ 68s] -- Could NOT find Git (missing: GIT_EXECUTABLE) [ 68s] -- Escargot Build Version: v3.0.0 [ 68s] -- Found PkgConfig: /bin/pkg-config (found version "0.28") [ 68s] -- Checking for module 'dlog' [ 68s] -- Found dlog, version 1.0 [ 68s] -- Configuring done [ 68s] -- Generating done [ 68s] -- Build files have been written to: /home/abuild/rpmbuild/BUILD/lwnode-1.0.0/out/tizen/Release/gen/escargot [ 68s] [75/231] CC obj/deps/brotli/c/enc/brotli.cluster.o [ 68s] [76/231] CC obj/deps/brotli/c/enc/brotli.backward_references_hq.o [ 68s] [77/231] CC obj/deps/brotli/c/enc/brotli.brotli_bit_stream.o [ 68s] [78/231] CC obj/deps/brotli/c/dec/brotli.decode.o [ 68s] [79/231] CC obj/deps/brotli/c/enc/brotli.block_splitter.o [ 69s] [80/231] CC obj/deps/brotli/c/enc/brotli.metablock.o [ 69s] [81/231] CC obj/deps/brotli/c/enc/brotli.encode.o [ 69s] [82/231] CC obj/deps/brotli/c/enc/brotli.compress_fragment_two_pass.o [ 70s] [83/231] CC obj/deps/brotli/c/enc/brotli.compress_fragment.o [ 75s] [84/231] CC obj/deps/brotli/c/enc/brotli.backward_references.o [ 75s] [85/231] AR obj/deps/brotli/libbrotli.a [ 120s] [86/231] ACTION escargot: build escargot_77aa8dffe56dc6e9e1faabd74ff0cddd [ 120s] ninja: Entering directory `/home/abuild/rpmbuild/BUILD/lwnode-1.0.0/out/tizen/Release/gen/escargot' [ 120s] [1/203] /bin/cc -I/home/abuild/rpmbuild/BUILD/lwnode-1.0.0/lwnode/code/escargotshim/deps/escargot/third_party/libbf -O2 -g2 -pipe -Wall -Wp,-D_FORTIFY_SOURCE=2 -fexceptions -fstack-protector-strong -Wformat-security -fmessage-length=0 -frecord-gcc-switches -Wl,-z,relro,--as-needed -feliminate-unused-debug-types --param=ssp-buffer-size=4 -fdiagnostics-color=never -m32 -march=i686 -mtune=i686 -msse4.2 -mfpmath=sse -fasynchronous-unwind-tables -fno-omit-frame-pointer -mstackrealign -Wl,-z,noseparate-code -g -Os -O2 -fno-stack-protector -m32 -fPIC -g2 -pipe -Wall -Wp,-D_FORTIFY_SOURCE=2 -fexceptions -fstack-protector-strong -Wformat-security -fmessage-length=0 -frecord-gcc-switches -Wl,-z,relro,--as-needed -feliminate-unused-debug-types --param=ssp-buffer-size=4 -fdiagnostics-color=never -march=i686 -mtune=i686 -msse4.2 -mfpmath=sse -fasynchronous-unwind-tables -fno-omit-frame-pointer -mstackrealign -Wl,-z,noseparate-code -g -Os -w -g3 -fdata-sections -ffunction-sections -fvisibility=hidden -MD -MT CMakeFiles/libbf.dir/third_party/libbf/cutils.c.o -MF CMakeFiles/libbf.dir/third_party/libbf/cutils.c.o.d -o CMakeFiles/libbf.dir/third_party/libbf/cutils.c.o -c /home/abuild/rpmbuild/BUILD/lwnode-1.0.0/lwnode/code/escargotshim/deps/escargot/third_party/libbf/cutils.c [ 120s] [2/203] /bin/c++ -DENABLE_COMPRESSIBLE_STRING -DENABLE_ICU -DENABLE_INTL -DENABLE_RELOADABLE_STRING -DENABLE_RUNTIME_ICU_BINDER -DESCARGOT -DESCARGOT_32=1 -DESCARGOT_SMALL_CONFIG -DESCARGOT_TIZEN -DLZ4_HEAPMODE=1 -DLZ4_MEMORY_USAGE=16 -DNDEBUG -Descargot_EXPORTS 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[ 120s] | ~~~~~ [ 120s] 91 | [ 120s] | [ 120s] 92 | typedef union { [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 93 | double value; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 94 | struct { [ 120s] | ~~~~~~~~ [ 120s] 95 | uint32_t msw; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 96 | uint32_t lsw; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 97 | } parts; [ 120s] | ~~~~~~~~ [ 120s] 98 | struct { [ 120s] | ~~~~~~~~ [ 120s] 99 | uint64_t w; [ 120s] | ~~~~~~~~~~~ [ 120s] 100 | } xparts; [ 120s] | ~~~~~~~~~ [ 120s] 101 | } ieee_double_shape_type; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 102 | [ 120s] | [ 120s] 103 | #define __HI(x) *(int *)&x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 104 | #define __LO(x) *(1 + (int *)&x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 105 | #define __HIp(x) *(int *)x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 106 | #define __LOp(x) *(1 + (int *)x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 107 | [ 120s] | [ 120s] 108 | #endif [ 120s] | ~~~~~~ [ 120s] 109 | [ 120s] | [ 120s] 110 | /* Get two 32 bit ints from a double. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 111 | [ 120s] | [ 120s] 112 | #define EXTRACT_WORDS(ix0, ix1, d) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 113 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 114 | ieee_double_shape_type ew_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 115 | ew_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 116 | (ix0) = ew_u.parts.msw; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 117 | (ix1) = ew_u.parts.lsw; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 118 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 119 | [ 120s] | [ 120s] 120 | /* Get a 64-bit int from a double. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 121 | #define EXTRACT_WORD64(ix, d) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 122 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 123 | ieee_double_shape_type ew_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 124 | ew_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 125 | (ix) = ew_u.xparts.w; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 126 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 127 | [ 120s] | [ 120s] 128 | /* Get the more significant 32 bit int from a double. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 129 | [ 120s] | [ 120s] 130 | #define GET_HIGH_WORD(i, d) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 131 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 132 | ieee_double_shape_type gh_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 133 | gh_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 134 | (i) = gh_u.parts.msw; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 135 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 136 | [ 120s] | [ 120s] 137 | /* Get the less significant 32 bit int from a double. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 138 | [ 120s] | [ 120s] 139 | #define GET_LOW_WORD(i, d) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 140 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 141 | ieee_double_shape_type gl_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 142 | gl_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 143 | (i) = gl_u.parts.lsw; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 144 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 145 | [ 120s] | [ 120s] 146 | /* Set a double from two 32 bit ints. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 147 | [ 120s] | [ 120s] 148 | #define INSERT_WORDS(d, ix0, ix1) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 149 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 150 | ieee_double_shape_type iw_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 151 | iw_u.parts.msw = (ix0); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 152 | iw_u.parts.lsw = (ix1); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 153 | (d) = iw_u.value; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 154 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 155 | [ 120s] | [ 120s] 156 | /* Set a double from a 64-bit int. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 157 | #define INSERT_WORD64(d, ix) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 158 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 159 | ieee_double_shape_type iw_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 160 | iw_u.xparts.w = (ix); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 161 | (d) = iw_u.value; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 162 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 163 | [ 120s] | [ 120s] 164 | /* Set the more significant 32 bits of a double from an int. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 165 | [ 120s] | [ 120s] 166 | #define SET_HIGH_WORD(d, v) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 167 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 168 | ieee_double_shape_type sh_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 169 | sh_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 170 | sh_u.parts.msw = (v); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 171 | (d) = sh_u.value; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 172 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 173 | [ 120s] | [ 120s] 174 | /* Set the less significant 32 bits of a double from an int. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 175 | [ 120s] | [ 120s] 176 | #define SET_LOW_WORD(d, v) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 177 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 178 | ieee_double_shape_type sl_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 179 | sl_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 180 | sl_u.parts.lsw = (v); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 181 | (d) = sl_u.value; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 182 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 183 | [ 120s] | [ 120s] 184 | /* Support macro. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 185 | [ 120s] | [ 120s] 186 | #define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 187 | [ 120s] | [ 120s] 188 | int32_t __ieee754_rem_pio2(double x, double *y) WARN_UNUSED_RETURN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 189 | double __kernel_cos(double x, double y) WARN_UNUSED_RETURN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 190 | int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 191 | const int32_t *ipio2) WARN_UNUSED_RETURN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 192 | double __kernel_sin(double x, double y, int iy) WARN_UNUSED_RETURN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 193 | [ 120s] | [ 120s] 194 | /* __ieee754_rem_pio2(x,y) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 195 | * [ 120s] | ~ [ 120s] 196 | * return the remainder of x rem pi/2 in y[0]+y[1] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 197 | * use __kernel_rem_pio2() [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 198 | */ [ 120s] | ~~ [ 120s] 199 | int32_t __ieee754_rem_pio2(double x, double *y) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 200 | { [ 120s] | ~ [ 120s] 201 | /* [ 120s] | ~~ [ 120s] 202 | * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 203 | */ [ 120s] | ~~ [ 120s] 204 | constexpr int32_t two_over_pi[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 205 | 0xA2F983, [ 120s] | ~~~~~~~~~ [ 120s] 206 | 0x6E4E44, [ 120s] | ~~~~~~~~~ [ 120s] 207 | 0x1529FC, [ 120s] | ~~~~~~~~~ [ 120s] 208 | 0x2757D1, [ 120s] | ~~~~~~~~~ [ 120s] 209 | 0xF534DD, [ 120s] | ~~~~~~~~~ [ 120s] 210 | 0xC0DB62, [ 120s] | ~~~~~~~~~ [ 120s] 211 | 0x95993C, [ 120s] | ~~~~~~~~~ [ 120s] 212 | 0x439041, [ 120s] | ~~~~~~~~~ [ 120s] 213 | 0xFE5163, [ 120s] | ~~~~~~~~~ [ 120s] 214 | 0xABDEBB, [ 120s] | ~~~~~~~~~ [ 120s] 215 | 0xC561B7, [ 120s] | ~~~~~~~~~ [ 120s] 216 | 0x246E3A, [ 120s] | ~~~~~~~~~ [ 120s] 217 | 0x424DD2, [ 120s] | ~~~~~~~~~ [ 120s] 218 | 0xE00649, [ 120s] | ~~~~~~~~~ [ 120s] 219 | 0x2EEA09, [ 120s] | ~~~~~~~~~ [ 120s] 220 | 0xD1921C, [ 120s] | ~~~~~~~~~ [ 120s] 221 | 0xFE1DEB, [ 120s] | ~~~~~~~~~ [ 120s] 222 | 0x1CB129, [ 120s] | ~~~~~~~~~ [ 120s] 223 | 0xA73EE8, [ 120s] | ~~~~~~~~~ [ 120s] 224 | 0x8235F5, [ 120s] | ~~~~~~~~~ [ 120s] 225 | 0x2EBB44, [ 120s] | ~~~~~~~~~ [ 120s] 226 | 0x84E99C, [ 120s] | ~~~~~~~~~ [ 120s] 227 | 0x7026B4, [ 120s] | ~~~~~~~~~ [ 120s] 228 | 0x5F7E41, [ 120s] | ~~~~~~~~~ [ 120s] 229 | 0x3991D6, [ 120s] | ~~~~~~~~~ [ 120s] 230 | 0x398353, [ 120s] | ~~~~~~~~~ [ 120s] 231 | 0x39F49C, [ 120s] | ~~~~~~~~~ [ 120s] 232 | 0x845F8B, [ 120s] | ~~~~~~~~~ [ 120s] 233 | 0xBDF928, [ 120s] | ~~~~~~~~~ [ 120s] 234 | 0x3B1FF8, [ 120s] | ~~~~~~~~~ [ 120s] 235 | 0x97FFDE, [ 120s] | ~~~~~~~~~ [ 120s] 236 | 0x05980F, [ 120s] | ~~~~~~~~~ [ 120s] 237 | 0xEF2F11, [ 120s] | ~~~~~~~~~ [ 120s] 238 | 0x8B5A0A, [ 120s] | ~~~~~~~~~ [ 120s] 239 | 0x6D1F6D, [ 120s] | ~~~~~~~~~ [ 120s] 240 | 0x367ECF, [ 120s] | ~~~~~~~~~ [ 120s] 241 | 0x27CB09, [ 120s] | ~~~~~~~~~ [ 120s] 242 | 0xB74F46, [ 120s] | ~~~~~~~~~ [ 120s] 243 | 0x3F669E, [ 120s] | ~~~~~~~~~ [ 120s] 244 | 0x5FEA2D, [ 120s] | ~~~~~~~~~ [ 120s] 245 | 0x7527BA, [ 120s] | ~~~~~~~~~ [ 120s] 246 | 0xC7EBE5, [ 120s] | ~~~~~~~~~ [ 120s] 247 | 0xF17B3D, [ 120s] | ~~~~~~~~~ [ 120s] 248 | 0x0739F7, [ 120s] | ~~~~~~~~~ [ 120s] 249 | 0x8A5292, [ 120s] | ~~~~~~~~~ [ 120s] 250 | 0xEA6BFB, [ 120s] | ~~~~~~~~~ [ 120s] 251 | 0x5FB11F, [ 120s] | ~~~~~~~~~ [ 120s] 252 | 0x8D5D08, [ 120s] | ~~~~~~~~~ [ 120s] 253 | 0x560330, [ 120s] | ~~~~~~~~~ [ 120s] 254 | 0x46FC7B, [ 120s] | ~~~~~~~~~ [ 120s] 255 | 0x6BABF0, [ 120s] | ~~~~~~~~~ [ 120s] 256 | 0xCFBC20, [ 120s] | ~~~~~~~~~ [ 120s] 257 | 0x9AF436, [ 120s] | ~~~~~~~~~ [ 120s] 258 | 0x1DA9E3, [ 120s] | ~~~~~~~~~ [ 120s] 259 | 0x91615E, [ 120s] | ~~~~~~~~~ [ 120s] 260 | 0xE61B08, [ 120s] | ~~~~~~~~~ [ 120s] 261 | 0x659985, [ 120s] | ~~~~~~~~~ [ 120s] 262 | 0x5F14A0, [ 120s] | ~~~~~~~~~ [ 120s] 263 | 0x68408D, [ 120s] | ~~~~~~~~~ [ 120s] 264 | 0xFFD880, [ 120s] | ~~~~~~~~~ [ 120s] 265 | 0x4D7327, [ 120s] | ~~~~~~~~~ [ 120s] 266 | 0x310606, [ 120s] | ~~~~~~~~~ [ 120s] 267 | 0x1556CA, [ 120s] | ~~~~~~~~~ [ 120s] 268 | 0x73A8C9, [ 120s] | ~~~~~~~~~ [ 120s] 269 | 0x60E27B, [ 120s] | ~~~~~~~~~ [ 120s] 270 | 0xC08C6B, [ 120s] | ~~~~~~~~~ [ 120s] 271 | }; [ 120s] | ~~ [ 120s] 272 | [ 120s] | [ 120s] 273 | constexpr int32_t npio2_hw[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 274 | 0x3FF921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 275 | 0x400921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 276 | 0x4012D97C, [ 120s] | ~~~~~~~~~~~ [ 120s] 277 | 0x401921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 278 | 0x401F6A7A, [ 120s] | ~~~~~~~~~~~ [ 120s] 279 | 0x4022D97C, [ 120s] | ~~~~~~~~~~~ [ 120s] 280 | 0x4025FDBB, [ 120s] | ~~~~~~~~~~~ [ 120s] 281 | 0x402921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 282 | 0x402C463A, [ 120s] | ~~~~~~~~~~~ [ 120s] 283 | 0x402F6A7A, [ 120s] | ~~~~~~~~~~~ [ 120s] 284 | 0x4031475C, [ 120s] | ~~~~~~~~~~~ [ 120s] 285 | 0x4032D97C, [ 120s] | ~~~~~~~~~~~ [ 120s] 286 | 0x40346B9C, [ 120s] | ~~~~~~~~~~~ [ 120s] 287 | 0x4035FDBB, [ 120s] | ~~~~~~~~~~~ [ 120s] 288 | 0x40378FDB, [ 120s] | ~~~~~~~~~~~ [ 120s] 289 | 0x403921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 290 | 0x403AB41B, [ 120s] | ~~~~~~~~~~~ [ 120s] 291 | 0x403C463A, [ 120s] | ~~~~~~~~~~~ [ 120s] 292 | 0x403DD85A, [ 120s] | ~~~~~~~~~~~ [ 120s] 293 | 0x403F6A7A, [ 120s] | ~~~~~~~~~~~ [ 120s] 294 | 0x40407E4C, [ 120s] | ~~~~~~~~~~~ [ 120s] 295 | 0x4041475C, [ 120s] | ~~~~~~~~~~~ [ 120s] 296 | 0x4042106C, [ 120s] | ~~~~~~~~~~~ [ 120s] 297 | 0x4042D97C, [ 120s] | ~~~~~~~~~~~ [ 120s] 298 | 0x4043A28C, [ 120s] | ~~~~~~~~~~~ [ 120s] 299 | 0x40446B9C, [ 120s] | ~~~~~~~~~~~ [ 120s] 300 | 0x404534AC, [ 120s] | ~~~~~~~~~~~ [ 120s] 301 | 0x4045FDBB, [ 120s] | ~~~~~~~~~~~ [ 120s] 302 | 0x4046C6CB, [ 120s] | ~~~~~~~~~~~ [ 120s] 303 | 0x40478FDB, [ 120s] | ~~~~~~~~~~~ [ 120s] 304 | 0x404858EB, [ 120s] | ~~~~~~~~~~~ [ 120s] 305 | 0x404921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 306 | }; [ 120s] | ~~ [ 120s] 307 | [ 120s] | [ 120s] 308 | /* [ 120s] | ~~ [ 120s] 309 | * invpio2: 53 bits of 2/pi [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 310 | * pio2_1: first 33 bit of pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 311 | * pio2_1t: pi/2 - pio2_1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 312 | * pio2_2: second 33 bit of pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 313 | * pio2_2t: pi/2 - (pio2_1+pio2_2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 314 | * pio2_3: third 33 bit of pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 315 | * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 316 | */ [ 120s] | ~~ [ 120s] 317 | [ 120s] | [ 120s] 318 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 319 | zero [ 120s] | ~~~~ [ 120s] 320 | = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 321 | half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 322 | two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 323 | invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 324 | pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 325 | pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 326 | pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 327 | pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 328 | pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 329 | pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 330 | [ 120s] | [ 120s] 331 | double z, w, t, r, fn; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 332 | double tx[3]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 333 | int32_t e0, i, j, nx, n, ix, hx; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 334 | uint32_t low; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 335 | [ 120s] | [ 120s] 336 | z = 0; [ 120s] | ~~~~~~ [ 120s] 337 | GET_HIGH_WORD(hx, x); /* high word of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 338 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 339 | if (ix <= 0x3FE921FB) { /* |x| ~<= pi/4 , no need for reduction */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 340 | y[0] = x; [ 120s] | ~~~~~~~~~ [ 120s] 341 | y[1] = 0; [ 120s] | ~~~~~~~~~ [ 120s] 342 | return 0; [ 120s] | ~~~~~~~~~ [ 120s] 343 | } [ 120s] | ~ [ 120s] 344 | if (ix < 0x4002D97C) { /* |x| < 3pi/4, special case with n=+-1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 345 | if (hx > 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 346 | z = x - pio2_1; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 347 | if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 348 | y[0] = z - pio2_1t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 349 | y[1] = (z - y[0]) - pio2_1t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 350 | } else { /* near pi/2, use 33+33+53 bit pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 351 | z -= pio2_2; [ 120s] | ~~~~~~~~~~~~ [ 120s] 352 | y[0] = z - pio2_2t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 353 | y[1] = (z - y[0]) - pio2_2t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 354 | } [ 120s] | ~ [ 120s] 355 | return 1; [ 120s] | ~~~~~~~~~ [ 120s] 356 | } else { /* negative x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 357 | z = x + pio2_1; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 358 | if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 359 | y[0] = z + pio2_1t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 360 | y[1] = (z - y[0]) + pio2_1t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 361 | } else { /* near pi/2, use 33+33+53 bit pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 362 | z += pio2_2; [ 120s] | ~~~~~~~~~~~~ [ 120s] 363 | y[0] = z + pio2_2t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 364 | y[1] = (z - y[0]) + pio2_2t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 365 | } [ 120s] | ~ [ 120s] 366 | return -1; [ 120s] | ~~~~~~~~~~ [ 120s] 367 | } [ 120s] | ~ [ 120s] 368 | } [ 120s] | ~ [ 120s] 369 | if (ix <= 0x413921FB) { /* |x| ~<= 2^19*(pi/2), medium size */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 370 | t = fabs(x); [ 120s] | ~~~~~~~~~~~~ [ 120s] 371 | n = static_cast(t * invpio2 + half); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 372 | fn = static_cast(n); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 373 | r = t - fn * pio2_1; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 374 | w = fn * pio2_1t; /* 1st round good to 85 bit */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 375 | if (n < 32 && ix != npio2_hw[n - 1]) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 376 | y[0] = r - w; /* quick check no cancellation */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 377 | } else { [ 120s] | ~~~~~~~~ [ 120s] 378 | uint32_t high; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 379 | j = ix >> 20; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 380 | y[0] = r - w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 381 | GET_HIGH_WORD(high, y[0]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 382 | i = j - ((high >> 20) & 0x7FF); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 383 | if (i > 16) { /* 2nd iteration needed, good to 118 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 384 | t = r; [ 120s] | ~~~~~~ [ 120s] 385 | w = fn * pio2_2; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 386 | r = t - w; [ 120s] | ~~~~~~~~~~ [ 120s] 387 | w = fn * pio2_2t - ((t - r) - w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 388 | y[0] = r - w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 389 | GET_HIGH_WORD(high, y[0]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 390 | i = j - ((high >> 20) & 0x7FF); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 391 | if (i > 49) { /* 3rd iteration need, 151 bits acc */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 392 | t = r; /* will cover all possible cases */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 393 | w = fn * pio2_3; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 394 | r = t - w; [ 120s] | ~~~~~~~~~~ [ 120s] 395 | w = fn * pio2_3t - ((t - r) - w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 396 | y[0] = r - w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 397 | } [ 120s] | ~ [ 120s] 398 | } [ 120s] | ~ [ 120s] 399 | } [ 120s] | ~ [ 120s] 400 | y[1] = (r - y[0]) - w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 401 | if (hx < 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 402 | y[0] = -y[0]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 403 | y[1] = -y[1]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 404 | return -n; [ 120s] | ~~~~~~~~~~ [ 120s] 405 | } else { [ 120s] | ~~~~~~~~ [ 120s] 406 | return n; [ 120s] | ~~~~~~~~~ [ 120s] 407 | } [ 120s] | ~ [ 120s] 408 | } [ 120s] | ~ [ 120s] 409 | /* [ 120s] | ~~ [ 120s] 410 | * all other (large) arguments [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 411 | */ [ 120s] | ~~ [ 120s] 412 | if (ix >= 0x7FF00000) { /* x is inf or NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 413 | y[0] = y[1] = x - x; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 414 | return 0; [ 120s] | ~~~~~~~~~ [ 120s] 415 | } [ 120s] | ~ [ 120s] 416 | /* set z = scalbn(|x|,ilogb(x)-23) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 417 | GET_LOW_WORD(low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 418 | SET_LOW_WORD(z, low); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 419 | e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 420 | SET_HIGH_WORD(z, ix - static_cast(e0 << 20)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 421 | for (i = 0; i < 2; i++) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 422 | tx[i] = static_cast(static_cast(z)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 423 | z = (z - tx[i]) * two24; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 424 | } [ 120s] | ~ [ 120s] 425 | tx[2] = z; [ 120s] | ~~~~~~~~~~ [ 120s] 426 | nx = 3; [ 120s] | ~~~~~~~ [ 120s] 427 | while (tx[nx - 1] == zero) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 428 | nx--; /* skip zero term */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 429 | n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 430 | if (hx < 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 431 | y[0] = -y[0]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 432 | y[1] = -y[1]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 433 | return -n; [ 120s] | ~~~~~~~~~~ [ 120s] 434 | } [ 120s] | ~ [ 120s] 435 | return n; [ 120s] | ~~~~~~~~~ [ 120s] 436 | } [ 120s] | ~ [ 120s] 437 | [ 120s] | [ 120s] 438 | /* __kernel_cos( x, y ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 439 | * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 440 | * Input x is assumed to be bounded by ~pi/4 in magnitude. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 441 | * Input y is the tail of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 442 | * [ 120s] | ~ [ 120s] 443 | * Algorithm [ 120s] | ~~~~~~~~~~~ [ 120s] 444 | * 1. Since cos(-x) = cos(x), we need only to consider positive x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 445 | * 2. if x < 2^-27 (hx<0x3E400000 0), return 1 with inexact if x!=0. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 446 | * 3. cos(x) is approximated by a polynomial of degree 14 on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 447 | * [0,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 448 | * 4 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 449 | * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 450 | * where the remez error is [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 451 | * [ 120s] | ~ [ 120s] 452 | * | 2 4 6 8 10 12 14 | -58 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 453 | * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 454 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 455 | * [ 120s] | ~ [ 120s] 456 | * 4 6 8 10 12 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 457 | * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 458 | * cos(x) = 1 - x*x/2 + r [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 459 | * since cos(x+y) ~ cos(x) - sin(x)*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 460 | * ~ cos(x) - x*y, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 461 | * a correction term is necessary in cos(x) and hence [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 462 | * cos(x+y) = 1 - (x*x/2 - (r - x*y)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 463 | * For better accuracy when x > 0.3, let qx = |x|/4 with [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 464 | * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 465 | * Then [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 466 | * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 467 | * Note that 1-qx and (x*x/2-qx) is EXACT here, and the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 468 | * magnitude of the latter is at least a quarter of x*x/2, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 469 | * thus, reducing the rounding error in the subtraction. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 470 | */ [ 120s] | ~~ [ 120s] 471 | ALWAYS_INLINE double __kernel_cos(double x, double y) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 472 | { [ 120s] | ~ [ 120s] 473 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 474 | one [ 120s] | ~~~ [ 120s] 475 | = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 476 | C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 477 | C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 478 | C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 479 | C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 480 | C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 481 | C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 482 | [ 120s] | [ 120s] 483 | double a, iz, z, r, qx; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 484 | int32_t ix; [ 120s] | ~~~~~~~~~~~ [ 120s] 485 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 486 | ix &= 0x7FFFFFFF; /* ix = |x|'s high word*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 487 | if (ix < 0x3E400000 && static_cast(x) == 0) { /* if x < 2**27 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 488 | return one; /* generate inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 489 | } [ 120s] | ~ [ 120s] 490 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 491 | r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 492 | if (ix < 0x3FD33333) { /* if |x| < 0.3 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 493 | return one - (0.5 * z - (z * r - x * y)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 494 | } else { [ 120s] | ~~~~~~~~ [ 120s] 495 | if (ix > 0x3FE90000) { /* x > 0.78125 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 496 | qx = 0.28125; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 497 | } else { [ 120s] | ~~~~~~~~ [ 120s] 498 | INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 499 | } [ 120s] | ~ [ 120s] 500 | iz = 0.5 * z - qx; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 501 | a = one - qx; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 502 | return a - (iz - (z * r - x * y)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 503 | } [ 120s] | ~ [ 120s] 504 | } [ 120s] | ~ [ 120s] 505 | [ 120s] | [ 120s] 506 | /* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 507 | * double x[],y[]; int e0,nx,prec; int ipio2[]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 508 | * [ 120s] | ~ [ 120s] 509 | * __kernel_rem_pio2 return the last three digits of N with [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 510 | * y = x - N*pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 511 | * so that |y| < pi/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 512 | * [ 120s] | ~ [ 120s] 513 | * The method is to compute the integer (mod 8) and fraction parts of [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 514 | * (2/pi)*x without doing the full multiplication. In general we [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 515 | * skip the part of the product that are known to be a huge integer ( [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 516 | * more accurately, = 0 mod 8 ). Thus the number of operations are [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 517 | * independent of the exponent of the input. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 518 | * [ 120s] | ~ [ 120s] 519 | * (2/pi) is represented by an array of 24-bit integers in ipio2[]. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 520 | * [ 120s] | ~ [ 120s] 521 | * Input parameters: [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 522 | * x[] The input value (must be positive) is broken into nx [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 523 | * pieces of 24-bit integers in double precision format. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 524 | * x[i] will be the i-th 24 bit of x. The scaled exponent [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 525 | * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 526 | * match x's up to 24 bits. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 527 | * [ 120s] | ~ [ 120s] 528 | * Example of breaking a double positive z into x[0]+x[1]+x[2]: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 529 | * e0 = ilogb(z)-23 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 530 | * z = scalbn(z,-e0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 531 | * for i = 0,1,2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 532 | * x[i] = floor(z) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 533 | * z = (z-x[i])*2**24 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 534 | * [ 120s] | ~ [ 120s] 535 | * [ 120s] | ~ [ 120s] 536 | * y[] output result in an array of double precision numbers. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 537 | * The dimension of y[] is: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 538 | * 24-bit precision 1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 539 | * 53-bit precision 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 540 | * 64-bit precision 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 541 | * 113-bit precision 3 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 542 | * The actual value is the sum of them. Thus for 113-bit [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 543 | * precison, one may have to do something like: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 544 | * [ 120s] | ~ [ 120s] 545 | * long double t,w,r_head, r_tail; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 546 | * t = (long double)y[2] + (long double)y[1]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 547 | * w = (long double)y[0]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 548 | * r_head = t+w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 549 | * r_tail = w - (r_head - t); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 550 | * [ 120s] | ~ [ 120s] 551 | * e0 The exponent of x[0] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 552 | * [ 120s] | ~ [ 120s] 553 | * nx dimension of x[] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 554 | * [ 120s] | ~ [ 120s] 555 | * prec an integer indicating the precision: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 556 | * 0 24 bits (single) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 557 | * 1 53 bits (double) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 558 | * 2 64 bits (extended) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 559 | * 3 113 bits (quad) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 560 | * [ 120s] | ~ [ 120s] 561 | * ipio2[] [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 562 | * integer array, contains the (24*i)-th to (24*i+23)-th [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 563 | * bit of 2/pi after binary point. The corresponding [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 564 | * floating value is [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 565 | * [ 120s] | ~ [ 120s] 566 | * ipio2[i] * 2^(-24(i+1)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 567 | * [ 120s] | ~ [ 120s] 568 | * External function: [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 569 | * double scalbn(), floor(); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 570 | * [ 120s] | ~ [ 120s] 571 | * [ 120s] | ~ [ 120s] 572 | * Here is the description of some local variables: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 573 | * [ 120s] | ~ [ 120s] 574 | * jk jk+1 is the initial number of terms of ipio2[] needed [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 575 | * in the computation. The recommended value is 2,3,4, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 576 | * 6 for single, double, extended,and quad. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 577 | * [ 120s] | ~ [ 120s] 578 | * jz local integer variable indicating the number of [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 579 | * terms of ipio2[] used. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 580 | * [ 120s] | ~ [ 120s] 581 | * jx nx - 1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 582 | * [ 120s] | ~ [ 120s] 583 | * jv index for pointing to the suitable ipio2[] for the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 584 | * computation. In general, we want [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 585 | * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 586 | * is an integer. Thus [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 587 | * e0-3-24*jv >= 0 or (e0-3)/24 >= jv [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 588 | * Hence jv = max(0,(e0-3)/24). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 589 | * [ 120s] | ~ [ 120s] 590 | * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 591 | * [ 120s] | ~ [ 120s] 592 | * q[] double array with integral value, representing the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 593 | * 24-bits chunk of the product of x and 2/pi. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 594 | * [ 120s] | ~ [ 120s] 595 | * q0 the corresponding exponent of q[0]. Note that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 596 | * exponent for q[i] would be q0-24*i. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 597 | * [ 120s] | ~ [ 120s] 598 | * PIo2[] double precision array, obtained by cutting pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 599 | * into 24 bits chunks. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 600 | * [ 120s] | ~ [ 120s] 601 | * f[] ipio2[] in floating point [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 602 | * [ 120s] | ~ [ 120s] 603 | * iq[] integer array by breaking up q[] in 24-bits chunk. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 604 | * [ 120s] | ~ [ 120s] 605 | * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 606 | * [ 120s] | ~ [ 120s] 607 | * ih integer. If >0 it indicates q[] is >= 0.5, hence [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 608 | * it also indicates the *sign* of the result. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 609 | * [ 120s] | ~ [ 120s] 610 | */ [ 120s] | ~~ [ 120s] 611 | int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 612 | const int32_t *ipio2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 613 | { [ 120s] | ~ [ 120s] 614 | /* Constants: [ 120s] | ~~~~~~~~~~~~~ [ 120s] 615 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 616 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 617 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 618 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 619 | */ [ 120s] | ~~ [ 120s] 620 | constexpr int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 621 | [ 120s] | [ 120s] 622 | constexpr double PIo2[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 623 | 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 624 | 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 625 | 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 626 | 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 627 | 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 628 | 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 629 | 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 630 | 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 631 | }; [ 120s] | ~~ [ 120s] 632 | [ 120s] | [ 120s] 633 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 634 | zero [ 120s] | ~~~~ [ 120s] 635 | = 0.0, [ 120s] | ~~~~~~ [ 120s] 636 | one = 1.0, [ 120s] | ~~~~~~~~~~ [ 120s] 637 | two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 638 | twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 639 | [ 120s] | [ 120s] 640 | int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 641 | double z, fw, f[20], fq[20], q[20]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 642 | [ 120s] | [ 120s] 643 | /* initialize jk*/ [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 644 | jk = init_jk[prec]; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 645 | jp = jk; [ 120s] | ~~~~~~~~ [ 120s] 646 | [ 120s] | [ 120s] 647 | /* determine jx,jv,q0, note that 3>q0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 648 | jx = nx - 1; [ 120s] | ~~~~~~~~~~~~ [ 120s] 649 | jv = (e0 - 3) / 24; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 650 | if (jv < 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 651 | jv = 0; [ 120s] | ~~~~~~~ [ 120s] 652 | q0 = e0 - 24 * (jv + 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 653 | [ 120s] | [ 120s] 654 | /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 655 | j = jv - jx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 656 | m = jx + jk; [ 120s] | ~~~~~~~~~~~~ [ 120s] 657 | for (i = 0; i <= m; i++, j++) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 658 | f[i] = (j < 0) ? zero : static_cast(ipio2[j]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 659 | } [ 120s] | ~ [ 120s] 660 | [ 120s] | [ 120s] 661 | /* compute q[0],q[1],...q[jk] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 662 | for (i = 0; i <= jk; i++) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 663 | for (j = 0, fw = 0.0; j <= jx; j++) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 664 | fw += x[j] * f[jx + i - j]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 665 | q[i] = fw; [ 120s] | ~~~~~~~~~~ [ 120s] 666 | } [ 120s] | ~ [ 120s] 667 | [ 120s] | [ 120s] 668 | jz = jk; [ 120s] | ~~~~~~~~ [ 120s] 669 | recompute: [ 120s] | ~~~~~~~~~~ [ 120s] 670 | /* distill q[] into iq[] reversingly */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 671 | for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 672 | fw = static_cast(static_cast(twon24 * z)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 673 | iq[i] = static_cast(z - two24 * fw); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 674 | z = q[j - 1] + fw; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 675 | } [ 120s] | ~ [ 120s] 676 | [ 120s] | [ 120s] 677 | /* compute n */ [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 678 | z = scalbn(z, q0); /* actual value of z */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 679 | z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 680 | n = static_cast(z); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 681 | z -= static_cast(n); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 682 | ih = 0; [ 120s] | ~~~~~~~ [ 120s] 683 | if (q0 > 0) { /* need iq[jz-1] to determine n */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 684 | i = (iq[jz - 1] >> (24 - q0)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 685 | n += i; [ 120s] | ~~~~~~~ [ 120s] 686 | iq[jz - 1] -= i << (24 - q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 687 | ih = iq[jz - 1] >> (23 - q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 688 | } else if (q0 == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 689 | ih = iq[jz - 1] >> 23; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 690 | } else if (z >= 0.5) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 691 | ih = 2; [ 120s] | ~~~~~~~ [ 120s] 692 | } [ 120s] | ~ [ 120s] 693 | [ 120s] | [ 120s] 694 | if (ih > 0) { /* q > 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 695 | n += 1; [ 120s] | ~~~~~~~ [ 120s] 696 | carry = 0; [ 120s] | ~~~~~~~~~~ [ 120s] 697 | for (i = 0; i < jz; i++) { /* compute 1-q */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 698 | j = iq[i]; [ 120s] | ~~~~~~~~~~ [ 120s] 699 | if (carry == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 700 | if (j != 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 701 | carry = 1; [ 120s] | ~~~~~~~~~~ [ 120s] 702 | iq[i] = 0x1000000 - j; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 703 | } [ 120s] | ~ [ 120s] 704 | } else { [ 120s] | ~~~~~~~~ [ 120s] 705 | iq[i] = 0xFFFFFF - j; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 706 | } [ 120s] | ~ [ 120s] 707 | } [ 120s] | ~ [ 120s] 708 | if (q0 > 0) { /* rare case: chance is 1 in 12 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 709 | switch (q0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 710 | case 1: [ 120s] | ~~~~~~~ [ 120s] 711 | iq[jz - 1] &= 0x7FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 712 | break; [ 120s] | ~~~~~~ [ 120s] 713 | case 2: [ 120s] | ~~~~~~~ [ 120s] 714 | iq[jz - 1] &= 0x3FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 715 | break; [ 120s] | ~~~~~~ [ 120s] 716 | } [ 120s] | ~ [ 120s] 717 | } [ 120s] | ~ [ 120s] 718 | if (ih == 2) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 719 | z = one - z; [ 120s] | ~~~~~~~~~~~~ [ 120s] 720 | if (carry != 0) [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 721 | z -= scalbn(one, q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 722 | } [ 120s] | ~ [ 120s] 723 | } [ 120s] | ~ [ 120s] 724 | [ 120s] | [ 120s] 725 | /* check if recomputation is needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 726 | if (z == zero) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 727 | j = 0; [ 120s] | ~~~~~~ [ 120s] 728 | for (i = jz - 1; i >= jk; i--) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 729 | j |= iq[i]; [ 120s] | ~~~~~~~~~~~ [ 120s] 730 | if (j == 0) { /* need recomputation */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 731 | for (k = 1; jk >= k && iq[jk - k] == 0; k++) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 732 | /* k = no. of terms needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 733 | } [ 120s] | ~ [ 120s] 734 | [ 120s] | [ 120s] 735 | for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 736 | f[jx + i] = ipio2[jv + i]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 737 | for (j = 0, fw = 0.0; j <= jx; j++) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 738 | fw += x[j] * f[jx + i - j]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 739 | q[i] = fw; [ 120s] | ~~~~~~~~~~ [ 120s] 740 | } [ 120s] | ~ [ 120s] 741 | jz += k; [ 120s] | ~~~~~~~~ [ 120s] 742 | goto recompute; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 743 | } [ 120s] | ~ [ 120s] 744 | } [ 120s] | ~ [ 120s] 745 | [ 120s] | [ 120s] 746 | /* chop off zero terms */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 747 | if (z == 0.0) { [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 748 | jz -= 1; [ 120s] | ~~~~~~~~ [ 120s] 749 | q0 -= 24; [ 120s] | ~~~~~~~~~ [ 120s] 750 | while (iq[jz] == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 751 | jz--; [ 120s] | ~~~~~ [ 120s] 752 | q0 -= 24; [ 120s] | ~~~~~~~~~ [ 120s] 753 | } [ 120s] | ~ [ 120s] 754 | } else { /* break z into 24-bit if necessary */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 755 | z = scalbn(z, -q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 756 | if (z >= two24) { [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 757 | fw = static_cast(static_cast(twon24 * z)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 758 | iq[jz] = z - two24 * fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 759 | jz += 1; [ 120s] | ~~~~~~~~ [ 120s] 760 | q0 += 24; [ 120s] | ~~~~~~~~~ [ 120s] 761 | iq[jz] = fw; [ 120s] | ~~~~~~~~~~~~ [ 120s] 762 | } else { [ 120s] | ~~~~~~~~ [ 120s] 763 | iq[jz] = z; [ 120s] | ~~~~~~~~~~~ [ 120s] 764 | } [ 120s] | ~ [ 120s] 765 | } [ 120s] | ~ [ 120s] 766 | [ 120s] | [ 120s] 767 | /* convert integer "bit" chunk to floating-point value */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 768 | fw = scalbn(one, q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 769 | for (i = jz; i >= 0; i--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 770 | q[i] = fw * iq[i]; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 771 | fw *= twon24; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 772 | } [ 120s] | ~ [ 120s] 773 | [ 120s] | [ 120s] 774 | /* compute PIo2[0,...,jp]*q[jz,...,0] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 775 | for (i = jz; i >= 0; i--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 776 | for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 777 | fw += PIo2[k] * q[i + k]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 778 | fq[jz - i] = fw; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 779 | } [ 120s] | ~ [ 120s] 780 | [ 120s] | [ 120s] 781 | /* compress fq[] into y[] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 782 | switch (prec) { [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 783 | case 0: [ 120s] | ~~~~~~~ [ 120s] 784 | fw = 0.0; [ 120s] | ~~~~~~~~~ [ 120s] 785 | for (i = jz; i >= 0; i--) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 786 | fw += fq[i]; [ 120s] | ~~~~~~~~~~~~ [ 120s] 787 | y[0] = (ih == 0) ? fw : -fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 788 | break; [ 120s] | ~~~~~~ [ 120s] 789 | case 1: [ 120s] | ~~~~~~~ [ 120s] 790 | case 2: [ 120s] | ~~~~~~~ [ 120s] 791 | fw = 0.0; [ 120s] | ~~~~~~~~~ [ 120s] 792 | for (i = jz; i >= 0; i--) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 793 | fw += fq[i]; [ 120s] | ~~~~~~~~~~~~ [ 120s] 794 | y[0] = (ih == 0) ? fw : -fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 795 | fw = fq[0] - fw; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 796 | for (i = 1; i <= jz; i++) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 797 | fw += fq[i]; [ 120s] | ~~~~~~~~~~~~ [ 120s] 798 | y[1] = (ih == 0) ? fw : -fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 799 | break; [ 120s] | ~~~~~~ [ 120s] 800 | case 3: /* painful */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 801 | for (i = jz; i > 0; i--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 802 | fw = fq[i - 1] + fq[i]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 803 | fq[i] += fq[i - 1] - fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 804 | fq[i - 1] = fw; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 805 | } [ 120s] | ~ [ 120s] 806 | for (i = jz; i > 1; i--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 807 | fw = fq[i - 1] + fq[i]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 808 | fq[i] += fq[i - 1] - fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 809 | fq[i - 1] = fw; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 810 | } [ 120s] | ~ [ 120s] 811 | for (fw = 0.0, i = jz; i >= 2; i--) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 812 | fw += fq[i]; [ 120s] | ~~~~~~~~~~~~ [ 120s] 813 | if (ih == 0) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 814 | y[0] = fq[0]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 815 | y[1] = fq[1]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 816 | y[2] = fw; [ 120s] | ~~~~~~~~~~ [ 120s] 817 | } else { [ 120s] | ~~~~~~~~ [ 120s] 818 | y[0] = -fq[0]; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 819 | y[1] = -fq[1]; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 820 | y[2] = -fw; [ 120s] | ~~~~~~~~~~~ [ 120s] 821 | } [ 120s] | ~ [ 120s] 822 | } [ 120s] | ~ [ 120s] 823 | return n & 7; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 824 | } [ 120s] | ~ [ 120s] 825 | [ 120s] | [ 120s] 826 | /* __kernel_sin( x, y, iy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 827 | * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 828 | * Input x is assumed to be bounded by ~pi/4 in magnitude. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 829 | * Input y is the tail of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 830 | * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 831 | * [ 120s] | ~ [ 120s] 832 | * Algorithm [ 120s] | ~~~~~~~~~~~ [ 120s] 833 | * 1. Since sin(-x) = -sin(x), we need only to consider positive x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 834 | * 2. if x < 2^-27 (hx<0x3E400000 0), return x with inexact if x!=0. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 835 | * 3. sin(x) is approximated by a polynomial of degree 13 on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 836 | * [0,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 837 | * 3 13 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 838 | * sin(x) ~ x + S1*x + ... + S6*x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 839 | * where [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 840 | * [ 120s] | ~ [ 120s] 841 | * |sin(x) 2 4 6 8 10 12 | -58 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 842 | * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 843 | * | x | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 844 | * [ 120s] | ~ [ 120s] 845 | * 4. sin(x+y) = sin(x) + sin'(x')*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 846 | * ~ sin(x) + (1-x*x/2)*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 847 | * For better accuracy, let [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 848 | * 3 2 2 2 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 849 | * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 850 | * then 3 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 851 | * sin(x) = x + (S1*x + (x *(r-y/2)+y)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 852 | */ [ 120s] | ~~ [ 120s] 853 | ALWAYS_INLINE double __kernel_sin(double x, double y, int iy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 854 | { [ 120s] | ~ [ 120s] 855 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 856 | half [ 120s] | ~~~~ [ 120s] 857 | = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 858 | S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 859 | S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 860 | S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 861 | S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 862 | S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 863 | S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 864 | [ 120s] | [ 120s] 865 | double z, r, v; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 866 | int32_t ix; [ 120s] | ~~~~~~~~~~~ [ 120s] 867 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 868 | ix &= 0x7FFFFFFF; /* high word of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 869 | if (ix < 0x3E400000 && static_cast(x) == 0) { /* |x| < 2**-27 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 870 | return x; [ 120s] | ~~~~~~~~~ [ 120s] 871 | } /* generate inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 872 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 873 | v = z * x; [ 120s] | ~~~~~~~~~~ [ 120s] 874 | r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 875 | if (iy == 0) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 876 | return x + v * (S1 + z * r); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 877 | } else { [ 120s] | ~~~~~~~~ [ 120s] 878 | return x - ((z * (half * y - v * r) - y) - v * S1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 879 | } [ 120s] | ~ [ 120s] 880 | } [ 120s] | ~ [ 120s] 881 | [ 120s] | [ 120s] 882 | /* __kernel_tan( x, y, k ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 883 | * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 884 | * Input x is assumed to be bounded by ~pi/4 in magnitude. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 885 | * Input y is the tail of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 886 | * Input k indicates whether tan (if k=1) or [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 887 | * -1/tan (if k= -1) is returned. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 888 | * [ 120s] | ~ [ 120s] 889 | * Algorithm [ 120s] | ~~~~~~~~~~~ [ 120s] 890 | * 1. Since tan(-x) = -tan(x), we need only to consider positive x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 891 | * 2. if x < 2^-28 (hx<0x3E300000 0), return x with inexact if x!=0. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 892 | * 3. tan(x) is approximated by a odd polynomial of degree 27 on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 893 | * [0,0.67434] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 894 | * 3 27 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 895 | * tan(x) ~ x + T1*x + ... + T13*x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 896 | * where [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 897 | * [ 120s] | ~ [ 120s] 898 | * |tan(x) 2 4 26 | -59.2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 899 | * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 900 | * | x | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 901 | * [ 120s] | ~ [ 120s] 902 | * Note: tan(x+y) = tan(x) + tan'(x)*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 903 | * ~ tan(x) + (1+x*x)*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 904 | * Therefore, for better accuracy in computing tan(x+y), let [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 905 | * 3 2 2 2 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 906 | * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 907 | * then [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 908 | * 3 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 909 | * tan(x+y) = x + (T1*x + (x *(r+y)+y)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 910 | * [ 120s] | ~ [ 120s] 911 | * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 912 | * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 913 | * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 914 | */ [ 120s] | ~~ [ 120s] 915 | double __kernel_tan(double x, double y, int iy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 916 | { [ 120s] | ~ [ 120s] 917 | constexpr double xxx[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 918 | 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 919 | 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 920 | 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 921 | 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 922 | 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 923 | 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 924 | 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 925 | 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 926 | 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 927 | 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 928 | 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 929 | -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 930 | 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 931 | /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 932 | /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 933 | /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 934 | }; [ 120s] | ~~ [ 120s] 935 | #define one xxx[13] [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 936 | #define pio4 xxx[14] [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 937 | #define pio4lo xxx[15] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 938 | #define T xxx [ 120s] | ~~~~~~~~~~~~~ [ 120s] 939 | [ 120s] | [ 120s] 940 | double z, r, v, w, s; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 941 | int32_t ix, hx; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 942 | [ 120s] | [ 120s] 943 | GET_HIGH_WORD(hx, x); /* high word of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 944 | ix = hx & 0x7FFFFFFF; /* high word of |x| */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 945 | if (ix < 0x3E300000 && static_cast(x) == 0) { /* x < 2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 946 | /* generate inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 947 | uint32_t low; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 948 | GET_LOW_WORD(low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 949 | if (((ix | low) | (iy + 1)) == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 950 | return one / fabs(x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 951 | } else { [ 120s] | ~~~~~~~~ [ 120s] 952 | if (iy == 1) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 953 | return x; [ 120s] | ~~~~~~~~~ [ 120s] 954 | } else { /* compute -1 / (x+y) carefully */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 955 | double a, t; [ 120s] | ~~~~~~~~~~~~ [ 120s] 956 | [ 120s] | [ 120s] 957 | z = w = x + y; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 958 | SET_LOW_WORD(z, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 959 | v = y - (z - x); [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 960 | t = a = -one / w; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 961 | SET_LOW_WORD(t, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 962 | s = one + t * z; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 963 | return t + a * (s + t * v); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 964 | } [ 120s] | ~ [ 120s] 965 | } [ 120s] | ~ [ 120s] 966 | } [ 120s] | ~ [ 120s] 967 | if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 968 | if (hx < 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 969 | x = -x; [ 120s] | ~~~~~~~ [ 120s] 970 | y = -y; [ 120s] | ~~~~~~~ [ 120s] 971 | } [ 120s] | ~ [ 120s] 972 | z = pio4 - x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 973 | w = pio4lo - y; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 974 | x = z + w; [ 120s] | ~~~~~~~~~~ [ 120s] 975 | y = 0.0; [ 120s] | ~~~~~~~~ [ 120s] 976 | } [ 120s] | ~ [ 120s] 977 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 978 | w = z * z; [ 120s] | ~~~~~~~~~~ [ 120s] 979 | /* [ 120s] | ~~ [ 120s] 980 | * Break x^5*(T[1]+x^2*T[2]+...) into [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 981 | * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 982 | * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 983 | */ [ 120s] | ~~ [ 120s] 984 | r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11])))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 985 | v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12]))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 986 | s = z * x; [ 120s] | ~~~~~~~~~~ [ 120s] 987 | r = y + z * (s * (r + v) + y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 988 | r += T[0] * s; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 989 | w = x + r; [ 120s] | ~~~~~~~~~~ [ 120s] 990 | if (ix >= 0x3FE59428) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 991 | v = iy; [ 120s] | ~~~~~~~ [ 120s] 992 | return (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 993 | } [ 120s] | ~ [ 120s] 994 | if (iy == 1) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 995 | return w; [ 120s] | ~~~~~~~~~ [ 120s] 996 | } else { [ 120s] | ~~~~~~~~ [ 120s] 997 | /* [ 120s] | ~~ [ 120s] 998 | * if allow error up to 2 ulp, simply return [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 999 | * -1.0 / (x+r) here [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1000 | */ [ 120s] | ~~ [ 120s] 1001 | /* compute -1.0 / (x+r) accurately */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1002 | double a, t; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1003 | z = w; [ 120s] | ~~~~~~ [ 120s] 1004 | SET_LOW_WORD(z, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1005 | v = r - (z - x); /* z+v = r+x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1006 | t = a = -1.0 / w; /* a = -1.0/w */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1007 | SET_LOW_WORD(t, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1008 | s = 1.0 + t * z; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1009 | return t + a * (s + t * v); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1010 | } [ 120s] | ~ [ 120s] 1011 | [ 120s] | [ 120s] 1012 | #undef one [ 120s] | ~~~~~~~~~~ [ 120s] 1013 | #undef pio4 [ 120s] | ~~~~~~~~~~~ [ 120s] 1014 | #undef pio4lo [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1015 | #undef T [ 120s] | ~~~~~~~~ [ 120s] 1016 | } [ 120s] | ~ [ 120s] 1017 | [ 120s] | [ 120s] 1018 | } // namespace [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1019 | [ 120s] | [ 120s] 1020 | /* acos(x) [ 120s] | ~~~~~~~~~~ [ 120s] 1021 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1022 | * acos(x) = pi/2 - asin(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1023 | * acos(-x) = pi/2 + asin(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1024 | * For |x|<=0.5 [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1025 | * acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1026 | * For x>0.5 [ 120s] | ~~~~~~~~~~~ [ 120s] 1027 | * acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1028 | * = 2asin(sqrt((1-x)/2)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1029 | * = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1030 | * = 2f + (2c + 2s*z*R(z)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1031 | * where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1032 | * for f so that f+c ~ sqrt(z). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1033 | * For x<-0.5 [ 120s] | ~~~~~~~~~~~~ [ 120s] 1034 | * acos(x) = pi - 2asin(sqrt((1-|x|)/2)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1035 | * = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1036 | * [ 120s] | ~ [ 120s] 1037 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1038 | * if x is NaN, return x itself; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1039 | * if |x|>1, return NaN with invalid signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1040 | * [ 120s] | ~ [ 120s] 1041 | * Function needed: sqrt [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1042 | */ [ 120s] | ~~ [ 120s] 1043 | double acos(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1044 | { [ 120s] | ~ [ 120s] 1045 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1046 | one [ 120s] | ~~~ [ 120s] 1047 | = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1048 | pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1049 | pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1050 | pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1051 | pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1052 | pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1053 | pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1054 | pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1055 | pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1056 | pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1057 | qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1058 | qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1059 | qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1060 | qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1061 | [ 120s] | [ 120s] 1062 | double z, p, q, r, w, s, c, df; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1063 | int32_t hx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1064 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1065 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1066 | if (ix >= 0x3FF00000) { /* |x| >= 1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1067 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1068 | GET_LOW_WORD(lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1069 | if (((ix - 0x3FF00000) | lx) == 0) { /* |x|==1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1070 | if (hx > 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1071 | return 0.0; /* acos(1) = 0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1072 | else [ 120s] | ~~~~ [ 120s] 1073 | return pi + 2.0 * pio2_lo; /* acos(-1)= pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1074 | } [ 120s] | ~ [ 120s] 1075 | return (x - x) / (x - x); /* acos(|x|>1) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1076 | } [ 120s] | ~ [ 120s] 1077 | if (ix < 0x3FE00000) { /* |x| < 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1078 | if (ix <= 0x3C600000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1079 | return pio2_hi + pio2_lo; /*if|x|<2**-57*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1080 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1081 | p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1082 | q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1083 | r = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1084 | return pio2_hi - (x - (pio2_lo - x * r)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1085 | } else if (hx < 0) { /* x < -0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1086 | z = (one + x) * 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1087 | p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1088 | q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1089 | s = sqrt(z); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1090 | r = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1091 | w = r * s - pio2_lo; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1092 | return pi - 2.0 * (s + w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1093 | } else { /* x > 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1094 | z = (one - x) * 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1095 | s = sqrt(z); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1096 | df = s; [ 120s] | ~~~~~~~ [ 120s] 1097 | SET_LOW_WORD(df, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1098 | c = (z - df * df) / (s + df); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1099 | p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1100 | q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1101 | r = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1102 | w = r * s + c; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1103 | return 2.0 * (df + w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1104 | } [ 120s] | ~ [ 120s] 1105 | } [ 120s] | ~ [ 120s] 1106 | [ 120s] | [ 120s] 1107 | /* acosh(x) [ 120s] | ~~~~~~~~~~~ [ 120s] 1108 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1109 | * Based on [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1110 | * acosh(x) = log [ x + sqrt(x*x-1) ] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1111 | * we have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1112 | * acosh(x) := log(x)+ln2, if x is large; else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1113 | * acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1114 | * acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1115 | * [ 120s] | ~ [ 120s] 1116 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1117 | * acosh(x) is NaN with signal if x<1. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1118 | * acosh(NaN) is NaN without signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1119 | */ [ 120s] | ~~ [ 120s] 1120 | double acosh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1121 | { [ 120s] | ~ [ 120s] 1122 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1123 | one [ 120s] | ~~~ [ 120s] 1124 | = 1.0, [ 120s] | ~~~~~~ [ 120s] 1125 | ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1126 | double t; [ 120s] | ~~~~~~~~~ [ 120s] 1127 | int32_t hx; [ 120s] | ~~~~~~~~~~~ [ 120s] 1128 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1129 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1130 | if (hx < 0x3FF00000) { /* x < 1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1131 | return (x - x) / (x - x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1132 | } else if (hx >= 0x41B00000) { /* x > 2**28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1133 | if (hx >= 0x7FF00000) { /* x is inf of NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1134 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1135 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1136 | return log(x) + ln2; /* acosh(huge)=log(2x) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1137 | } [ 120s] | ~ [ 120s] 1138 | } else if (((hx - 0x3FF00000) | lx) == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1139 | return 0.0; /* acosh(1) = 0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1140 | } else if (hx > 0x40000000) { /* 2**28 > x > 2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1141 | t = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1142 | return log(2.0 * x - one / (x + sqrt(t - one))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1143 | } else { /* 10.98 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1163 | * asin(x) = pi/2 - 2*(s+s*z*R(z)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1164 | * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1165 | * For x<=0.98, let pio4_hi = pio2_hi/2, then [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1166 | * f = hi part of s; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1167 | * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1168 | * and [ 120s] | ~~~~~~~~~~ [ 120s] 1169 | * asin(x) = pi/2 - 2*(s+s*z*R(z)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1170 | * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1171 | * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1172 | * [ 120s] | ~ [ 120s] 1173 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1174 | * if x is NaN, return x itself; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1175 | * if |x|>1, return NaN with invalid signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1176 | */ [ 120s] | ~~ [ 120s] 1177 | double asin(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1178 | { [ 120s] | ~ [ 120s] 1179 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1180 | one [ 120s] | ~~~ [ 120s] 1181 | = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1182 | huge = 1.000e+300, [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1183 | pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1184 | pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1185 | pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1186 | /* coefficient for R(x^2) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1187 | pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1188 | pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1189 | pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1190 | pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1191 | pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1192 | pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1193 | qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1194 | qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1195 | qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1196 | qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1197 | [ 120s] | [ 120s] 1198 | double t, w, p, q, c, r, s; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1199 | int32_t hx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1200 | [ 120s] | [ 120s] 1201 | t = 0; [ 120s] | ~~~~~~ [ 120s] 1202 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1203 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1204 | if (ix >= 0x3FF00000) { /* |x|>= 1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1205 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1206 | GET_LOW_WORD(lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1207 | if (((ix - 0x3FF00000) | lx) == 0) /* asin(1)=+-pi/2 with inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1208 | return x * pio2_hi + x * pio2_lo; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1209 | return (x - x) / (x - x); /* asin(|x|>1) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1210 | } else if (ix < 0x3FE00000) { /* |x|<0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1211 | if (ix < 0x3E400000 && huge + x > one) { /* if |x| < 2**-27 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1212 | return x; /* return x with inexact if x!=0*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1213 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1214 | t = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1215 | } [ 120s] | ~ [ 120s] 1216 | p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1217 | q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1218 | w = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1219 | return x + x * w; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1220 | } [ 120s] | ~ [ 120s] 1221 | /* 1> |x|>= 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1222 | w = one - fabs(x); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1223 | t = w * 0.5; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1224 | p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1225 | q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1226 | s = sqrt(t); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1227 | if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1228 | w = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1229 | t = pio2_hi - (2.0 * (s + s * w) - pio2_lo); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1230 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1231 | w = s; [ 120s] | ~~~~~~ [ 120s] 1232 | SET_LOW_WORD(w, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1233 | c = (t - w * w) / (s + w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1234 | r = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1235 | p = 2.0 * s * r - (pio2_lo - 2.0 * c); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1236 | q = pio4_hi - 2.0 * w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1237 | t = pio4_hi - (p - q); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1238 | } [ 120s] | ~ [ 120s] 1239 | if (hx > 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1240 | return t; [ 120s] | ~~~~~~~~~ [ 120s] 1241 | else [ 120s] | ~~~~ [ 120s] 1242 | return -t; [ 120s] | ~~~~~~~~~~ [ 120s] 1243 | } [ 120s] | ~ [ 120s] 1244 | /* asinh(x) [ 120s] | ~~~~~~~~~~~ [ 120s] 1245 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1246 | * Based on [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1247 | * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1248 | * we have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1249 | * asinh(x) := x if 1+x*x=1, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1250 | * := sign(x)*(log(x)+ln2)) for large |x|, else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1251 | * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1252 | * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1253 | */ [ 120s] | ~~ [ 120s] 1254 | double asinh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1255 | { [ 120s] | ~ [ 120s] 1256 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1257 | one [ 120s] | ~~~ [ 120s] 1258 | = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1259 | ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1260 | huge = 1.00000000000000000000e+300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1261 | [ 120s] | [ 120s] 1262 | double t, w; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1263 | int32_t hx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1264 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1265 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1266 | if (ix >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1267 | return x + x; /* x is inf or NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1268 | if (ix < 0x3E300000 && huge + x > one) { /* |x|<2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1269 | return x; /* return x inexact except 0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1270 | } [ 120s] | ~ [ 120s] 1271 | if (ix > 0x41B00000) { /* |x| > 2**28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1272 | w = log(fabs(x)) + ln2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1273 | } else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1274 | t = fabs(x); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1275 | w = log(2.0 * t + one / (sqrt(x * x + one) + t)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1276 | } else { /* 2.0 > |x| > 2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1277 | t = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1278 | w = log1p(fabs(x) + t / (one + sqrt(one + t))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1279 | } [ 120s] | ~ [ 120s] 1280 | if (hx > 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1281 | return w; [ 120s] | ~~~~~~~~~ [ 120s] 1282 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1283 | return -w; [ 120s] | ~~~~~~~~~~ [ 120s] 1284 | } [ 120s] | ~ [ 120s] 1285 | } [ 120s] | ~ [ 120s] 1286 | [ 120s] | [ 120s] 1287 | /* atan(x) [ 120s] | ~~~~~~~~~~ [ 120s] 1288 | * Method [ 120s] | ~~~~~~~~ [ 120s] 1289 | * 1. Reduce x to positive by atan(x) = -atan(-x). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1290 | * 2. According to the integer k=4t+0.25 chopped, t=x, the argument [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1291 | * is further reduced to one of the following intervals and the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1292 | * arctangent of t is evaluated by the corresponding formula: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1293 | * [ 120s] | ~ [ 120s] 1294 | * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1295 | * [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1296 | * [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1297 | * [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1298 | * [39/16,INF] atan(x) = atan(INF) + atan( -1/t ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1299 | * [ 120s] | ~ [ 120s] 1300 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1301 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1302 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1303 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1304 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1305 | */ [ 120s] | ~~ [ 120s] 1306 | double atan(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1307 | { [ 120s] | ~ [ 120s] 1308 | constexpr double atanhi[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1309 | 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1310 | 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1311 | 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1312 | 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1313 | }; [ 120s] | ~~ [ 120s] 1314 | [ 120s] | [ 120s] 1315 | constexpr double atanlo[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1316 | 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1317 | 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1318 | 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1319 | 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1320 | }; [ 120s] | ~~ [ 120s] 1321 | [ 120s] | [ 120s] 1322 | constexpr double aT[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1323 | 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1324 | -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1325 | 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1326 | -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1327 | 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1328 | -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1329 | 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1330 | -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1331 | 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1332 | -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1333 | 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1334 | }; [ 120s] | ~~ [ 120s] 1335 | [ 120s] | [ 120s] 1336 | constexpr double one = 1.0, huge = 1.0e300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1337 | [ 120s] | [ 120s] 1338 | double w, s1, s2, z; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1339 | int32_t ix, hx, id; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1340 | [ 120s] | [ 120s] 1341 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1342 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1343 | if (ix >= 0x44100000) { /* if |x| >= 2^66 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1344 | uint32_t low; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1345 | GET_LOW_WORD(low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1346 | if (ix > 0x7FF00000 || (ix == 0x7FF00000 && (low != 0))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1347 | return x + x; /* NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1348 | if (hx > 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1349 | return atanhi[3] + *(volatile double *)&atanlo[3]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1350 | else [ 120s] | ~~~~ [ 120s] 1351 | return -atanhi[3] - *(volatile double *)&atanlo[3]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1352 | } [ 120s] | ~ [ 120s] 1353 | if (ix < 0x3FDC0000) { /* |x| < 0.4375 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1354 | if (ix < 0x3E400000 && huge + x > one) { /* |x| < 2^-27 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1355 | return x; /* raise inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1356 | } [ 120s] | ~ [ 120s] 1357 | id = -1; [ 120s] | ~~~~~~~~ [ 120s] 1358 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1359 | x = fabs(x); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1360 | if (ix < 0x3FF30000) { /* |x| < 1.1875 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1361 | if (ix < 0x3FE60000) { /* 7/16 <=|x|<11/16 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1362 | id = 0; [ 120s] | ~~~~~~~ [ 120s] 1363 | x = (2.0 * x - one) / (2.0 + x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1364 | } else { /* 11/16<=|x|< 19/16 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1365 | id = 1; [ 120s] | ~~~~~~~ [ 120s] 1366 | x = (x - one) / (x + one); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1367 | } [ 120s] | ~ [ 120s] 1368 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1369 | if (ix < 0x40038000) { /* |x| < 2.4375 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1370 | id = 2; [ 120s] | ~~~~~~~ [ 120s] 1371 | x = (x - 1.5) / (one + 1.5 * x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1372 | } else { /* 2.4375 <= |x| < 2^66 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1373 | id = 3; [ 120s] | ~~~~~~~ [ 120s] 1374 | x = -1.0 / x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1375 | } [ 120s] | ~ [ 120s] 1376 | } [ 120s] | ~ [ 120s] 1377 | } [ 120s] | ~ [ 120s] 1378 | /* end of argument reduction */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1379 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1380 | w = z * z; [ 120s] | ~~~~~~~~~~ [ 120s] 1381 | /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1382 | s1 = z * (aT[0] + w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10]))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1383 | s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9])))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1384 | if (id < 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1385 | return x - x * (s1 + s2); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1386 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1387 | z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1388 | return (hx < 0) ? -z : z; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1389 | } [ 120s] | ~ [ 120s] 1390 | } [ 120s] | ~ [ 120s] 1391 | [ 120s] | [ 120s] 1392 | /* atan2(y,x) [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1393 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1394 | * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1395 | * 2. Reduce x to positive by (if x and y are unexceptional): [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1396 | * ARG (x+iy) = arctan(y/x) ... if x > 0, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1397 | * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1398 | * [ 120s] | ~ [ 120s] 1399 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1400 | * [ 120s] | ~ [ 120s] 1401 | * ATAN2((anything), NaN ) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1402 | * ATAN2(NAN , (anything) ) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1403 | * ATAN2(+-0, +(anything but NaN)) is +-0 ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1404 | * ATAN2(+-0, -(anything but NaN)) is +-pi ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1405 | * ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1406 | * ATAN2(+-(anything but INF and NaN), +INF) is +-0 ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1407 | * ATAN2(+-(anything but INF and NaN), -INF) is +-pi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1408 | * ATAN2(+-INF,+INF ) is +-pi/4 ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1409 | * ATAN2(+-INF,-INF ) is +-3pi/4; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1410 | * ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1411 | * [ 120s] | ~ [ 120s] 1412 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1413 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1414 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1415 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1416 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1417 | */ [ 120s] | ~~ [ 120s] 1418 | double atan2(double y, double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1419 | { [ 120s] | ~ [ 120s] 1420 | constexpr volatile double tiny = 1.0e-300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1421 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1422 | zero [ 120s] | ~~~~ [ 120s] 1423 | = 0.0, [ 120s] | ~~~~~~ [ 120s] 1424 | pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1425 | pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1426 | pi = 3.1415926535897931160E+00; /* 0x400921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1427 | constexpr volatile double pi_lo = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1428 | [ 120s] | [ 120s] 1429 | double z; [ 120s] | ~~~~~~~~~ [ 120s] 1430 | int32_t k, m, hx, hy, ix, iy; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1431 | uint32_t lx, ly; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1432 | [ 120s] | [ 120s] 1433 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1434 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1435 | EXTRACT_WORDS(hy, ly, y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1436 | iy = hy & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1437 | if (((ix | ((lx | -static_cast(lx)) >> 31)) > 0x7FF00000) || ((iy | ((ly | -static_cast(ly)) >> 31)) > 0x7FF00000)) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1438 | return x + y; /* x or y is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1439 | } [ 120s] | ~ [ 120s] 1440 | if (((hx - 0x3FF00000) | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1441 | return atan(y); /* x=1.0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1442 | m = ((hy >> 31) & 1) | ((hx >> 30) & 2); /* 2*sign(x)+sign(y) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1443 | [ 120s] | [ 120s] 1444 | /* when y = 0 */ [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1445 | if ((iy | ly) == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1446 | switch (m) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1447 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1448 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1449 | return y; /* atan(+-0,+anything)=+-0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1450 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1451 | return pi + tiny; /* atan(+0,-anything) = pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1452 | case 3: [ 120s] | ~~~~~~~ [ 120s] 1453 | return -pi - tiny; /* atan(-0,-anything) =-pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1454 | } [ 120s] | ~ [ 120s] 1455 | } [ 120s] | ~ [ 120s] 1456 | /* when x = 0 */ [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1457 | if ((ix | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1458 | return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1459 | [ 120s] | [ 120s] 1460 | /* when x is INF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1461 | if (ix == 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1462 | if (iy == 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1463 | switch (m) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1464 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1465 | return pi_o_4 + tiny; /* atan(+INF,+INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1466 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1467 | return -pi_o_4 - tiny; /* atan(-INF,+INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1468 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1469 | return 3.0 * pi_o_4 + tiny; /*atan(+INF,-INF)*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1470 | case 3: [ 120s] | ~~~~~~~ [ 120s] 1471 | return -3.0 * pi_o_4 - tiny; /*atan(-INF,-INF)*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1472 | } [ 120s] | ~ [ 120s] 1473 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1474 | switch (m) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1475 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1476 | return zero; /* atan(+...,+INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1477 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1478 | return -zero; /* atan(-...,+INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1479 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1480 | return pi + tiny; /* atan(+...,-INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1481 | case 3: [ 120s] | ~~~~~~~ [ 120s] 1482 | return -pi - tiny; /* atan(-...,-INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1483 | } [ 120s] | ~ [ 120s] 1484 | } [ 120s] | ~ [ 120s] 1485 | } [ 120s] | ~ [ 120s] 1486 | /* when y is INF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1487 | if (iy == 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1488 | return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1489 | [ 120s] | [ 120s] 1490 | /* compute y/x */ [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1491 | k = (iy - ix) >> 20; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1492 | if (k > 60) { /* |y/x| > 2**60 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1493 | z = pi_o_2 + 0.5 * pi_lo; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1494 | m &= 1; [ 120s] | ~~~~~~~ [ 120s] 1495 | } else if (hx < 0 && k < -60) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1496 | z = 0.0; /* 0 > |y|/x > -2**-60 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1497 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1498 | z = atan(fabs(y / x)); /* safe to do y/x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1499 | } [ 120s] | ~ [ 120s] 1500 | switch (m) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1501 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1502 | return z; /* atan(+,+) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1503 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1504 | return -z; /* atan(-,+) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1505 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1506 | return pi - (z - pi_lo); /* atan(+,-) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1507 | default: /* case 3 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1508 | return (z - pi_lo) - pi; /* atan(-,-) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1509 | } [ 120s] | ~ [ 120s] 1510 | } [ 120s] | ~ [ 120s] 1511 | [ 120s] | [ 120s] 1512 | /* cos(x) [ 120s] | ~~~~~~~~~ [ 120s] 1513 | * Return cosine function of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1514 | * [ 120s] | ~ [ 120s] 1515 | * kernel function: [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1516 | * __kernel_sin ... sine function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1517 | * __kernel_cos ... cosine function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1518 | * __ieee754_rem_pio2 ... argument reduction routine [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1519 | * [ 120s] | ~ [ 120s] 1520 | * Method. [ 120s] | ~~~~~~~~~ [ 120s] 1521 | * Let S,C and T denote the sin, cos and tan respectively on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1522 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1523 | * in [-pi/4 , +pi/4], and let n = k mod 4. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1524 | * We have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1525 | * [ 120s] | ~ [ 120s] 1526 | * n sin(x) cos(x) tan(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1527 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1528 | * 0 S C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1529 | * 1 C -S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1530 | * 2 -S -C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1531 | * 3 -C S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1532 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1533 | * [ 120s] | ~ [ 120s] 1534 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1535 | * Let trig be any of sin, cos, or tan. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1536 | * trig(+-INF) is NaN, with signals; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1537 | * trig(NaN) is that NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1538 | * [ 120s] | ~ [ 120s] 1539 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 1540 | * TRIG(x) returns trig(x) nearly rounded [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1541 | */ [ 120s] | ~~ [ 120s] 1542 | double cos(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1543 | { [ 120s] | ~ [ 120s] 1544 | double y[2], z = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1545 | int32_t n, ix; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1546 | [ 120s] | [ 120s] 1547 | /* High word of x. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1548 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1549 | [ 120s] | [ 120s] 1550 | /* |x| ~< pi/4 */ [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1551 | ix &= 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1552 | if (ix <= 0x3FE921FB) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1553 | return __kernel_cos(x, z); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1554 | } else if (ix >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1555 | /* cos(Inf or NaN) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1556 | return x - x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1557 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1558 | /* argument reduction needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1559 | n = __ieee754_rem_pio2(x, y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1560 | switch (n & 3) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1561 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1562 | return __kernel_cos(y[0], y[1]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1563 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1564 | return -__kernel_sin(y[0], y[1], 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1565 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1566 | return -__kernel_cos(y[0], y[1]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1567 | default: [ 120s] | ~~~~~~~~ [ 120s] 1568 | return __kernel_sin(y[0], y[1], 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1569 | } [ 120s] | ~ [ 120s] 1570 | } [ 120s] | ~ [ 120s] 1571 | } [ 120s] | ~ [ 120s] 1572 | [ 120s] | [ 120s] 1573 | /* exp(x) [ 120s] | ~~~~~~~~~ [ 120s] 1574 | * Returns the exponential of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1575 | * [ 120s] | ~ [ 120s] 1576 | * Method [ 120s] | ~~~~~~~~ [ 120s] 1577 | * 1. Argument reduction: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1578 | * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1579 | * Given x, find r and integer k such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1580 | * [ 120s] | ~ [ 120s] 1581 | * x = k*ln2 + r, |r| <= 0.5*ln2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1582 | * [ 120s] | ~ [ 120s] 1583 | * Here r will be represented as r = hi-lo for better [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1584 | * accuracy. [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1585 | * [ 120s] | ~ [ 120s] 1586 | * 2. Approximation of exp(r) by a special rational function on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1587 | * the interval [0,0.34658]: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1588 | * Write [ 120s] | ~~~~~~~~~~~~ [ 120s] 1589 | * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1590 | * We use a special Remes algorithm on [0,0.34658] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1591 | * a polynomial of degree 5 to approximate R. The maximum error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1592 | * of this polynomial approximation is bounded by 2**-59. In [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1593 | * other words, [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1594 | * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1595 | * (where z=r*r, and the values of P1 to P5 are listed below) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1596 | * and [ 120s] | ~~~~~~~~~~ [ 120s] 1597 | * | 5 | -59 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1598 | * | 2.0+P1*z+...+P5*z - R(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1599 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1600 | * The computation of exp(r) thus becomes [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1601 | * 2*r [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1602 | * exp(r) = 1 + ------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1603 | * R - r [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1604 | * r*R1(r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1605 | * = 1 + r + ----------- (for better accuracy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1606 | * 2 - R1(r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1607 | * where [ 120s] | ~~~~~~~~~~~~ [ 120s] 1608 | * 2 4 10 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1609 | * R1(r) = r - (P1*r + P2*r + ... + P5*r ). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1610 | * [ 120s] | ~ [ 120s] 1611 | * 3. Scale back to obtain exp(x): [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1612 | * From step 1, we have [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1613 | * exp(x) = 2^k * exp(r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1614 | * [ 120s] | ~ [ 120s] 1615 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1616 | * exp(INF) is INF, exp(NaN) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1617 | * exp(-INF) is 0, and [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1618 | * for finite argument, only exp(0)=1 is exact. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1619 | * [ 120s] | ~ [ 120s] 1620 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 1621 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1622 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1623 | * [ 120s] | ~ [ 120s] 1624 | * Misc. info. [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1625 | * For IEEE double [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1626 | * if x > 7.09782712893383973096e+02 then exp(x) overflow [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1627 | * if x < -7.45133219101941108420e+02 then exp(x) underflow [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1628 | * [ 120s] | ~ [ 120s] 1629 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1630 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1631 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1632 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1633 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1634 | */ [ 120s] | ~~ [ 120s] 1635 | double exp(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1636 | { [ 120s] | ~ [ 120s] 1637 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1638 | one [ 120s] | ~~~ [ 120s] 1639 | = 1.0, [ 120s] | ~~~~~~ [ 120s] 1640 | halF[2] = { 0.5, -0.5 }, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1641 | o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1642 | u_threshold = -7.45133219101941108420e+02, /* 0xC0874910, 0xD52D3051 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1643 | ln2HI[2] = { 6.93147180369123816490e-01, /* 0x3FE62E42, 0xFEE00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1644 | -6.93147180369123816490e-01 }, /* 0xBFE62E42, 0xFEE00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1645 | ln2LO[2] = { 1.90821492927058770002e-10, /* 0x3DEA39EF, 0x35793C76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1646 | -1.90821492927058770002e-10 }, /* 0xBDEA39EF, 0x35793C76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1647 | invln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1648 | P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1649 | P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1650 | P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1651 | P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1652 | P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1653 | E = 2.718281828459045; /* 0x4005BF0A, 0x8B145769 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1654 | [ 120s] | [ 120s] 1655 | constexpr volatile double [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1656 | huge [ 120s] | ~~~~ [ 120s] 1657 | = 1.0e+300, [ 120s] | ~~~~~~~~~~~ [ 120s] 1658 | twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1659 | two1023 = 8.988465674311579539e307; /* 0x1p1023 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1660 | [ 120s] | [ 120s] 1661 | double y, hi = 0.0, lo = 0.0, c, t, twopk; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1662 | int32_t k = 0, xsb; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1663 | uint32_t hx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1664 | [ 120s] | [ 120s] 1665 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1666 | xsb = (hx >> 31) & 1; /* sign bit of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1667 | hx &= 0x7FFFFFFF; /* high word of |x| */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1668 | [ 120s] | [ 120s] 1669 | /* filter out non-finite argument */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1670 | if (hx >= 0x40862E42) { /* if |x|>=709.78... */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1671 | if (hx >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1672 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1673 | GET_LOW_WORD(lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1674 | if (((hx & 0xFFFFF) | lx) != 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1675 | return x + x; /* NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1676 | else [ 120s] | ~~~~ [ 120s] 1677 | return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1678 | } [ 120s] | ~ [ 120s] 1679 | if (x > o_threshold) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1680 | return huge * huge; /* overflow */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1681 | if (x < u_threshold) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1682 | return twom1000 * twom1000; /* underflow */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1683 | } [ 120s] | ~ [ 120s] 1684 | [ 120s] | [ 120s] 1685 | /* argument reduction */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1686 | if (hx > 0x3FD62E42) { /* if |x| > 0.5 ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1687 | if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1688 | /* TODO(rtoy): We special case exp(1) here to return the correct [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1689 | * value of E, as the computation below would get the last bit [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1690 | * wrong. We should probably fix the algorithm instead. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1691 | */ [ 120s] | ~~ [ 120s] 1692 | if (x == 1.0) [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1693 | return E; [ 120s] | ~~~~~~~~~ [ 120s] 1694 | hi = x - ln2HI[xsb]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1695 | lo = ln2LO[xsb]; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1696 | k = 1 - xsb - xsb; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1697 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1698 | k = static_cast(invln2 * x + halF[xsb]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1699 | t = k; [ 120s] | ~~~~~~ [ 120s] 1700 | hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1701 | lo = t * ln2LO[0]; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1702 | } [ 120s] | ~ [ 120s] 1703 | STRICT_ASSIGN(double, x, hi - lo); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1704 | } else if (hx < 0x3E300000) { /* when |x|<2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1705 | if (huge + x > one) [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1706 | return one + x; /* trigger inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1707 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1708 | k = 0; [ 120s] | ~~~~~~ [ 120s] 1709 | } [ 120s] | ~ [ 120s] 1710 | [ 120s] | [ 120s] 1711 | /* x is now in primary range */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1712 | t = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1713 | if (k >= -1021) { [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1714 | INSERT_WORDS(twopk, 0x3FF00000 + (k << 20), 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1715 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1716 | INSERT_WORDS(twopk, 0x3FF00000 + ((k + 1000) << 20), 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1717 | } [ 120s] | ~ [ 120s] 1718 | c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1719 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1720 | return one - ((x * c) / (c - 2.0) - x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1721 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1722 | y = one - ((lo - (x * c) / (2.0 - c)) - hi); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1723 | } [ 120s] | ~ [ 120s] 1724 | if (k >= -1021) { [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1725 | if (k == 1024) [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1726 | return y * 2.0 * two1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1727 | return y * twopk; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1728 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1729 | return y * twopk * twom1000; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1730 | } [ 120s] | ~ [ 120s] 1731 | } [ 120s] | ~ [ 120s] 1732 | [ 120s] | [ 120s] 1733 | /* [ 120s] | ~~ [ 120s] 1734 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1735 | * 1.Reduced x to positive by atanh(-x) = -atanh(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1736 | * 2.For x>=0.5 [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1737 | * 1 2x x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1738 | * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1739 | * 2 1 - x 1 - x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1740 | * [ 120s] | ~ [ 120s] 1741 | * For x<0.5 [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1742 | * atanh(x) = 0.5*log1p(2x+2x*x/(1-x)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1743 | * [ 120s] | ~ [ 120s] 1744 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1745 | * atanh(x) is NaN if |x| > 1 with signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1746 | * atanh(NaN) is that NaN with no signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1747 | * atanh(+-1) is +-INF with signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1748 | * [ 120s] | ~ [ 120s] 1749 | */ [ 120s] | ~~ [ 120s] 1750 | double atanh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1751 | { [ 120s] | ~ [ 120s] 1752 | constexpr double one = 1.0, huge = 1e300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1753 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1754 | [ 120s] | [ 120s] 1755 | double t; [ 120s] | ~~~~~~~~~ [ 120s] 1756 | int32_t hx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1757 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1758 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1759 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1760 | if ((ix | ((lx | -static_cast(lx)) >> 31)) > 0x3FF00000) /* |x|>1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1761 | return (x - x) / (x - x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1762 | if (ix == 0x3FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1763 | return x / zero; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1764 | if (ix < 0x3E300000 && (huge + x) > zero) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1765 | return x; /* x<2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1766 | SET_HIGH_WORD(x, ix); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1767 | if (ix < 0x3FE00000) { /* x < 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1768 | t = x + x; [ 120s] | ~~~~~~~~~~ [ 120s] 1769 | t = 0.5 * log1p(t + t * x / (one - x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1770 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1771 | t = 0.5 * log1p((x + x) / (one - x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1772 | } [ 120s] | ~ [ 120s] 1773 | if (hx >= 0) [ 120s] | ~~~~~~~~~~~~ [ 120s] 1774 | return t; [ 120s] | ~~~~~~~~~ [ 120s] 1775 | else [ 120s] | ~~~~ [ 120s] 1776 | return -t; [ 120s] | ~~~~~~~~~~ [ 120s] 1777 | } [ 120s] | ~ [ 120s] 1778 | [ 120s] | [ 120s] 1779 | /* log(x) [ 120s] | ~~~~~~~~~ [ 120s] 1780 | * Return the logrithm of x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1781 | * [ 120s] | ~ [ 120s] 1782 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1783 | * 1. Argument Reduction: find k and f such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1784 | * x = 2^k * (1+f), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1785 | * where sqrt(2)/2 < 1+f < sqrt(2) . [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1786 | * [ 120s] | ~ [ 120s] 1787 | * 2. Approximation of log(1+f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1788 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1789 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1790 | * = 2s + s*R [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1791 | * We use a special Reme algorithm on [0,0.1716] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1792 | * a polynomial of degree 14 to approximate R The maximum error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1793 | * of this polynomial approximation is bounded by 2**-58.45. In [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1794 | * other words, [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1795 | * 2 4 6 8 10 12 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1796 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1797 | * (the values of Lg1 to Lg7 are listed in the program) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1798 | * and [ 120s] | ~~~~~~ [ 120s] 1799 | * | 2 14 | -58.45 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1800 | * | Lg1*s +...+Lg7*s - R(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1801 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1802 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1803 | * In order to guarantee error in log below 1ulp, we compute log [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1804 | * by [ 120s] | ~~~~~ [ 120s] 1805 | * log(1+f) = f - s*(f - R) (if f is not too large) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1806 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1807 | * [ 120s] | ~ [ 120s] 1808 | * 3. Finally, log(x) = k*ln2 + log(1+f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1809 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1810 | * Here ln2 is split into two floating point number: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1811 | * ln2_hi + ln2_lo, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1812 | * where n*ln2_hi is always exact for |n| < 2000. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1813 | * [ 120s] | ~ [ 120s] 1814 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1815 | * log(x) is NaN with signal if x < 0 (including -INF) ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1816 | * log(+INF) is +INF; log(0) is -INF with signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1817 | * log(NaN) is that NaN with no signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1818 | * [ 120s] | ~ [ 120s] 1819 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 1820 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1821 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1822 | * [ 120s] | ~ [ 120s] 1823 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1824 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1825 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1826 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1827 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1828 | */ [ 120s] | ~~ [ 120s] 1829 | double log(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1830 | { [ 120s] | ~ [ 120s] 1831 | constexpr double /* -- */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1832 | ln2_hi [ 120s] | ~~~~~~ [ 120s] 1833 | = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1834 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1835 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1836 | Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1837 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1838 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1839 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1840 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1841 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1842 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1843 | [ 120s] | [ 120s] 1844 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1845 | constexpr volatile double vzero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1846 | [ 120s] | [ 120s] 1847 | double hfsq, f, s, z, R, w, t1, t2, dk; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1848 | int32_t k, hx, i, j; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1849 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1850 | [ 120s] | [ 120s] 1851 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1852 | [ 120s] | [ 120s] 1853 | k = 0; [ 120s] | ~~~~~~ [ 120s] 1854 | if (hx < 0x00100000) { /* x < 2**-1022 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1855 | if (((hx & 0x7FFFFFFF) | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1856 | return -two54 / vzero; /* log(+-0)=-inf */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1857 | if (hx < 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1858 | return (x - x) / zero; /* log(-#) = NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1859 | k -= 54; [ 120s] | ~~~~~~~~ [ 120s] 1860 | x *= two54; /* subnormal number, scale up x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1861 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1862 | } [ 120s] | ~ [ 120s] 1863 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1864 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1865 | k += (hx >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1866 | hx &= 0x000FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1867 | i = (hx + 0x95F64) & 0x100000; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1868 | SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1869 | k += (i >> 20); [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1870 | f = x - 1.0; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1871 | if ((0x000FFFFF & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1872 | if (f == zero) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1873 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1874 | return zero; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1875 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1876 | dk = static_cast(k); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1877 | return dk * ln2_hi + dk * ln2_lo; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1878 | } [ 120s] | ~ [ 120s] 1879 | } [ 120s] | ~ [ 120s] 1880 | R = f * f * (0.5 - 0.33333333333333333 * f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1881 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1882 | return f - R; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1883 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1884 | dk = static_cast(k); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1885 | return dk * ln2_hi - ((R - dk * ln2_lo) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1886 | } [ 120s] | ~ [ 120s] 1887 | } [ 120s] | ~ [ 120s] 1888 | s = f / (2.0 + f); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1889 | dk = static_cast(k); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1890 | z = s * s; [ 120s] | ~~~~~~~~~~ [ 120s] 1891 | i = hx - 0x6147A; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1892 | w = z * z; [ 120s] | ~~~~~~~~~~ [ 120s] 1893 | j = 0x6B851 - hx; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1894 | t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1895 | t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1896 | i |= j; [ 120s] | ~~~~~~~ [ 120s] 1897 | R = t2 + t1; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1898 | if (i > 0) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1899 | hfsq = 0.5 * f * f; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1900 | if (k == 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1901 | return f - (hfsq - s * (hfsq + R)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1902 | else [ 120s] | ~~~~ [ 120s] 1903 | return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1904 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1905 | if (k == 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1906 | return f - s * (f - R); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1907 | else [ 120s] | ~~~~ [ 120s] 1908 | return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1909 | } [ 120s] | ~ [ 120s] 1910 | } [ 120s] | ~ [ 120s] 1911 | [ 120s] | [ 120s] 1912 | /* double log1p(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1913 | * [ 120s] | ~ [ 120s] 1914 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1915 | * 1. Argument Reduction: find k and f such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1916 | * 1+x = 2^k * (1+f), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1917 | * where sqrt(2)/2 < 1+f < sqrt(2) . [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1918 | * [ 120s] | ~ [ 120s] 1919 | * Note. If k=0, then f=x is exact. However, if k!=0, then f [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1920 | * may not be representable exactly. In that case, a correction [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1921 | * term is need. Let u=1+x rounded. Let c = (1+x)-u, then [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1922 | * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1923 | * and add back the correction term c/u. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1924 | * (Note: when x > 2**53, one can simply return log(x)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1925 | * [ 120s] | ~ [ 120s] 1926 | * 2. Approximation of log1p(f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1927 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1928 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1929 | * = 2s + s*R [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1930 | * We use a special Reme algorithm on [0,0.1716] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1931 | * a polynomial of degree 14 to approximate R The maximum error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1932 | * of this polynomial approximation is bounded by 2**-58.45. In [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1933 | * other words, [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1934 | * 2 4 6 8 10 12 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1935 | * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1936 | * (the values of Lp1 to Lp7 are listed in the program) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1937 | * and [ 120s] | ~~~~~~ [ 120s] 1938 | * | 2 14 | -58.45 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1939 | * | Lp1*s +...+Lp7*s - R(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1940 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1941 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1942 | * In order to guarantee error in log below 1ulp, we compute log [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1943 | * by [ 120s] | ~~~~~ [ 120s] 1944 | * log1p(f) = f - (hfsq - s*(hfsq+R)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1945 | * [ 120s] | ~ [ 120s] 1946 | * 3. Finally, log1p(x) = k*ln2 + log1p(f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1947 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1948 | * Here ln2 is split into two floating point number: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1949 | * ln2_hi + ln2_lo, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1950 | * where n*ln2_hi is always exact for |n| < 2000. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1951 | * [ 120s] | ~ [ 120s] 1952 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1953 | * log1p(x) is NaN with signal if x < -1 (including -INF) ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1954 | * log1p(+INF) is +INF; log1p(-1) is -INF with signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1955 | * log1p(NaN) is that NaN with no signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1956 | * [ 120s] | ~ [ 120s] 1957 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 1958 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1959 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1960 | * [ 120s] | ~ [ 120s] 1961 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1962 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1963 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1964 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1965 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1966 | * [ 120s] | ~ [ 120s] 1967 | * Note: Assuming log() return accurate answer, the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1968 | * algorithm can be used to compute log1p(x) to within a few ULP: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1969 | * [ 120s] | ~ [ 120s] 1970 | * u = 1+x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1971 | * if(u==1.0) return x ; else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1972 | * return log(u)*(x/(u-1.0)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1973 | * [ 120s] | ~ [ 120s] 1974 | * See HP-15C Advanced Functions Handbook, p.193. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1975 | */ [ 120s] | ~~ [ 120s] 1976 | double log1p(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1977 | { [ 120s] | ~ [ 120s] 1978 | constexpr double /* -- */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1979 | ln2_hi [ 120s] | ~~~~~~ [ 120s] 1980 | = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1981 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1982 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1983 | Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1984 | Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1985 | Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1986 | Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1987 | Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1988 | Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1989 | Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1990 | [ 120s] | [ 120s] 1991 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1992 | constexpr volatile double vzero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1993 | [ 120s] | [ 120s] 1994 | double hfsq, f, c, s, z, R, u; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1995 | int32_t k, hx, hu, ax; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1996 | [ 120s] | [ 120s] 1997 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1998 | ax = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1999 | [ 120s] | [ 120s] 2000 | c = 0; [ 120s] | ~~~~~~ [ 120s] 2001 | k = 1; [ 120s] | ~~~~~~ [ 120s] 2002 | if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2003 | if (ax >= 0x3FF00000) { /* x <= -1.0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2004 | if (x == -1.0) [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2005 | return -two54 / vzero; /* log1p(-1)=+inf */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2006 | else [ 120s] | ~~~~ [ 120s] 2007 | return (x - x) / (x - x); /* log1p(x<-1)=NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2008 | } [ 120s] | ~ [ 120s] 2009 | if (ax < 0x3E200000) { /* |x| < 2**-29 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2010 | if (two54 + x > zero /* raise inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2011 | && ax < 0x3C900000) /* |x| < 2**-54 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2012 | return x; [ 120s] | ~~~~~~~~~ [ 120s] 2013 | else [ 120s] | ~~~~ [ 120s] 2014 | return x - x * x * 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2015 | } [ 120s] | ~ [ 120s] 2016 | if (hx > 0 || hx <= static_cast(0xBFD2BEC4)) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2017 | k = 0; [ 120s] | ~~~~~~ [ 120s] 2018 | f = x; [ 120s] | ~~~~~~ [ 120s] 2019 | hu = 1; [ 120s] | ~~~~~~~ [ 120s] 2020 | } /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2021 | } [ 120s] | ~ [ 120s] 2022 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2023 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2024 | if (k != 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2025 | if (hx < 0x43400000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2026 | STRICT_ASSIGN(double, u, 1.0 + x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2027 | GET_HIGH_WORD(hu, u); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2028 | k = (hu >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2029 | c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2030 | c /= u; [ 120s] | ~~~~~~~ [ 120s] 2031 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2032 | u = x; [ 120s] | ~~~~~~ [ 120s] 2033 | GET_HIGH_WORD(hu, u); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2034 | k = (hu >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2035 | c = 0; [ 120s] | ~~~~~~ [ 120s] 2036 | } [ 120s] | ~ [ 120s] 2037 | hu &= 0x000FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2038 | /* [ 120s] | ~~ [ 120s] 2039 | * The approximation to sqrt(2) used in thresholds is not [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2040 | * critical. However, the ones used above must give less [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2041 | * strict bounds than the one here so that the k==0 case is [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2042 | * never reached from here, since here we have committed to [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2043 | * using the correction term but don't use it if k==0. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2044 | */ [ 120s] | ~~ [ 120s] 2045 | if (hu < 0x6A09E) { /* u ~< sqrt(2) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2046 | SET_HIGH_WORD(u, hu | 0x3FF00000); /* normalize u */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2047 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2048 | k += 1; [ 120s] | ~~~~~~~ [ 120s] 2049 | SET_HIGH_WORD(u, hu | 0x3FE00000); /* normalize u/2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2050 | hu = (0x00100000 - hu) >> 2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2051 | } [ 120s] | ~ [ 120s] 2052 | f = u - 1.0; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2053 | } [ 120s] | ~ [ 120s] 2054 | hfsq = 0.5 * f * f; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2055 | if (hu == 0) { /* |f| < 2**-20 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2056 | if (f == zero) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2057 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2058 | return zero; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2059 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2060 | c += k * ln2_lo; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2061 | return k * ln2_hi + c; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2062 | } [ 120s] | ~ [ 120s] 2063 | } [ 120s] | ~ [ 120s] 2064 | R = hfsq * (1.0 - 0.66666666666666666 * f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2065 | if (k == 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 2066 | return f - R; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2067 | else [ 120s] | ~~~~ [ 120s] 2068 | return k * ln2_hi - ((R - (k * ln2_lo + c)) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2069 | } [ 120s] | ~ [ 120s] 2070 | s = f / (2.0 + f); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2071 | z = s * s; [ 120s] | ~~~~~~~~~~ [ 120s] 2072 | R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7)))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2073 | if (k == 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 2074 | return f - (hfsq - s * (hfsq + R)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2075 | else [ 120s] | ~~~~ [ 120s] 2076 | return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2077 | } [ 120s] | ~ [ 120s] 2078 | [ 120s] | [ 120s] 2079 | /* [ 120s] | ~~ [ 120s] 2080 | * k_log1p(f): [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2081 | * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)]. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2082 | * [ 120s] | ~ [ 120s] 2083 | * The following describes the overall strategy for computing [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2084 | * logarithms in base e. The argument reduction and adding the final [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2085 | * term of the polynomial are done by the caller for increased accuracy [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2086 | * when different bases are used. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2087 | * [ 120s] | ~ [ 120s] 2088 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2089 | * 1. Argument Reduction: find k and f such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2090 | * x = 2^k * (1+f), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2091 | * where sqrt(2)/2 < 1+f < sqrt(2) . [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2092 | * [ 120s] | ~ [ 120s] 2093 | * 2. Approximation of log(1+f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2094 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2095 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2096 | * = 2s + s*R [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2097 | * We use a special Reme algorithm on [0,0.1716] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2098 | * a polynomial of degree 14 to approximate R The maximum error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2099 | * of this polynomial approximation is bounded by 2**-58.45. In [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2100 | * other words, [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2101 | * 2 4 6 8 10 12 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2102 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2103 | * (the values of Lg1 to Lg7 are listed in the program) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2104 | * and [ 120s] | ~~~~~~~~~~ [ 120s] 2105 | * | 2 14 | -58.45 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2106 | * | Lg1*s +...+Lg7*s - R(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2107 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2108 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2109 | * In order to guarantee error in log below 1ulp, we compute log [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2110 | * by [ 120s] | ~~~~~~~~~ [ 120s] 2111 | * log(1+f) = f - s*(f - R) (if f is not too large) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2112 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2113 | * [ 120s] | ~ [ 120s] 2114 | * 3. Finally, log(x) = k*ln2 + log(1+f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2115 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2116 | * Here ln2 is split into two floating point number: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2117 | * ln2_hi + ln2_lo, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2118 | * where n*ln2_hi is always exact for |n| < 2000. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2119 | * [ 120s] | ~ [ 120s] 2120 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2121 | * log(x) is NaN with signal if x < 0 (including -INF) ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2122 | * log(+INF) is +INF; log(0) is -INF with signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2123 | * log(NaN) is that NaN with no signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2124 | * [ 120s] | ~ [ 120s] 2125 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 2126 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2127 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2128 | * [ 120s] | ~ [ 120s] 2129 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 2130 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2131 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2132 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2133 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2134 | */ [ 120s] | ~~ [ 120s] 2135 | [ 120s] | [ 120s] 2136 | constexpr double Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2137 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2138 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2139 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2140 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2141 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2142 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2143 | [ 120s] | [ 120s] 2144 | /* [ 120s] | ~~ [ 120s] 2145 | * We always inline k_log1p(), since doing so produces a [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2146 | * substantial performance improvement (~40% on amd64). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2147 | */ [ 120s] | ~~ [ 120s] 2148 | static inline double k_log1p(double f) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2149 | { [ 120s] | ~ [ 120s] 2150 | double hfsq, s, z, R, w, t1, t2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2151 | [ 120s] | [ 120s] 2152 | s = f / (2.0 + f); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2153 | z = s * s; [ 120s] | ~~~~~~~~~~ [ 120s] 2154 | w = z * z; [ 120s] | ~~~~~~~~~~ [ 120s] 2155 | t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2156 | t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2157 | R = t2 + t1; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2158 | hfsq = 0.5 * f * f; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2159 | return s * (hfsq + R); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2160 | } [ 120s] | ~ [ 120s] 2161 | [ 120s] | [ 120s] 2162 | /* [ 120s] | ~~ [ 120s] 2163 | * Return the base 2 logarithm of x. See e_log.c and k_log.h for most [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2164 | * comments. [ 120s] | ~~~~~~~~~~~ [ 120s] 2165 | * [ 120s] | ~ [ 120s] 2166 | * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2167 | * then does the combining and scaling steps [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2168 | * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2169 | * in not-quite-routine extra precision. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2170 | */ [ 120s] | ~~ [ 120s] 2171 | double log2(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2172 | { [ 120s] | ~ [ 120s] 2173 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2174 | two54 [ 120s] | ~~~~~ [ 120s] 2175 | = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2176 | ivln2hi = 1.44269504072144627571e+00, /* 0x3FF71547, 0x65200000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2177 | ivln2lo = 1.67517131648865118353e-10; /* 0x3DE705FC, 0x2EEFA200 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2178 | [ 120s] | [ 120s] 2179 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2180 | constexpr volatile double vzero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2181 | [ 120s] | [ 120s] 2182 | double f, hfsq, hi, lo, r, val_hi, val_lo, w, y; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2183 | int32_t i, k, hx; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2184 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2185 | [ 120s] | [ 120s] 2186 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2187 | [ 120s] | [ 120s] 2188 | k = 0; [ 120s] | ~~~~~~ [ 120s] 2189 | if (hx < 0x00100000) { /* x < 2**-1022 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2190 | if (((hx & 0x7FFFFFFF) | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2191 | return -two54 / vzero; /* log(+-0)=-inf */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2192 | if (hx < 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 2193 | return (x - x) / zero; /* log(-#) = NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2194 | k -= 54; [ 120s] | ~~~~~~~~ [ 120s] 2195 | x *= two54; /* subnormal number, scale up x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2196 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2197 | } [ 120s] | ~ [ 120s] 2198 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2199 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2200 | if (hx == 0x3FF00000 && lx == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2201 | return zero; /* log(1) = +0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2202 | k += (hx >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2203 | hx &= 0x000FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2204 | i = (hx + 0x95F64) & 0x100000; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2205 | SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2206 | k += (i >> 20); [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2207 | y = static_cast(k); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2208 | f = x - 1.0; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2209 | hfsq = 0.5 * f * f; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2210 | r = k_log1p(f); [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2211 | [ 120s] | [ 120s] 2212 | /* [ 120s] | ~~ [ 120s] 2213 | * f-hfsq must (for args near 1) be evaluated in extra precision [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2214 | * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2215 | * This is fairly efficient since f-hfsq only depends on f, so can [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2216 | * be evaluated in parallel with R. Not combining hfsq with R also [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2217 | * keeps R small (though not as small as a true `lo' term would be), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2218 | * so that extra precision is not needed for terms involving R. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2219 | * [ 120s] | ~ [ 120s] 2220 | * Compiler bugs involving extra precision used to break Dekker's [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2221 | * theorem for spitting f-hfsq as hi+lo, unless double_t was used [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2222 | * or the multi-precision calculations were avoided when double_t [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2223 | * has extra precision. These problems are now automatically [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2224 | * avoided as a side effect of the optimization of combining the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2225 | * Dekker splitting step with the clear-low-bits step. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2226 | * [ 120s] | ~ [ 120s] 2227 | * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2228 | * precision to avoid a very large cancellation when x is very near [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2229 | * these values. Unlike the above cancellations, this problem is [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2230 | * specific to base 2. It is strange that adding +-1 is so much [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2231 | * harder than adding +-ln2 or +-log10_2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2232 | * [ 120s] | ~ [ 120s] 2233 | * This uses Dekker's theorem to normalize y+val_hi, so the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2234 | * compiler bugs are back in some configurations, sigh. And I [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2235 | * don't want to used double_t to avoid them, since that gives a [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2236 | * pessimization and the support for avoiding the pessimization [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2237 | * is not yet available. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2238 | * [ 120s] | ~ [ 120s] 2239 | * The multi-precision calculations for the multiplications are [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2240 | * routine. [ 120s] | ~~~~~~~~~~ [ 120s] 2241 | */ [ 120s] | ~~ [ 120s] 2242 | hi = f - hfsq; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2243 | SET_LOW_WORD(hi, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2244 | lo = (f - hi) - hfsq + r; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2245 | val_hi = hi * ivln2hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2246 | val_lo = (lo + hi) * ivln2lo + lo * ivln2hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2247 | [ 120s] | [ 120s] 2248 | /* spadd(val_hi, val_lo, y), except for not using double_t: */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2249 | w = y + val_hi; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2250 | val_lo += (y - w) + val_hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2251 | val_hi = w; [ 120s] | ~~~~~~~~~~~ [ 120s] 2252 | [ 120s] | [ 120s] 2253 | return val_lo + val_hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2254 | } [ 120s] | ~ [ 120s] 2255 | [ 120s] | [ 120s] 2256 | /* [ 120s] | ~~ [ 120s] 2257 | * Return the base 10 logarithm of x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2258 | * [ 120s] | ~ [ 120s] 2259 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2260 | * Let log10_2hi = leading 40 bits of log10(2) and [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2261 | * log10_2lo = log10(2) - log10_2hi, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2262 | * ivln10 = 1/log(10) rounded. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2263 | * Then [ 120s] | ~~~~~~~~~~~ [ 120s] 2264 | * n = ilogb(x), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2265 | * if(n<0) n = n+1; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2266 | * x = scalbn(x,-n); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2267 | * log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2268 | * [ 120s] | ~ [ 120s] 2269 | * Note 1: [ 120s] | ~~~~~~~~~~ [ 120s] 2270 | * To guarantee log10(10**n)=n, where 10**n is normal, the rounding [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2271 | * mode must set to Round-to-Nearest. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2272 | * Note 2: [ 120s] | ~~~~~~~~~~ [ 120s] 2273 | * [1/log(10)] rounded to 53 bits has error .198 ulps; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2274 | * log10 is monotonic at all binary break points. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2275 | * [ 120s] | ~ [ 120s] 2276 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2277 | * log10(x) is NaN if x < 0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2278 | * log10(+INF) is +INF; log10(0) is -INF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2279 | * log10(NaN) is that NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2280 | * log10(10**N) = N for N=0,1,...,22. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2281 | */ [ 120s] | ~~ [ 120s] 2282 | double log10(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2283 | { [ 120s] | ~ [ 120s] 2284 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2285 | two54 [ 120s] | ~~~~~ [ 120s] 2286 | = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2287 | ivln10 = 4.34294481903251816668e-01, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2288 | log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2289 | log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2290 | [ 120s] | [ 120s] 2291 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2292 | constexpr volatile double vzero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2293 | [ 120s] | [ 120s] 2294 | double y; [ 120s] | ~~~~~~~~~ [ 120s] 2295 | int32_t i, k, hx; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2296 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2297 | [ 120s] | [ 120s] 2298 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2299 | [ 120s] | [ 120s] 2300 | k = 0; [ 120s] | ~~~~~~ [ 120s] 2301 | if (hx < 0x00100000) { /* x < 2**-1022 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2302 | if (((hx & 0x7FFFFFFF) | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2303 | return -two54 / vzero; /* log(+-0)=-inf */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2304 | if (hx < 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 2305 | return (x - x) / zero; /* log(-#) = NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2306 | k -= 54; [ 120s] | ~~~~~~~~ [ 120s] 2307 | x *= two54; /* subnormal number, scale up x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2308 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2309 | GET_LOW_WORD(lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2310 | } [ 120s] | ~ [ 120s] 2311 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2312 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2313 | if (hx == 0x3FF00000 && lx == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2314 | return zero; /* log(1) = +0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2315 | k += (hx >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2316 | [ 120s] | [ 120s] 2317 | i = (k & 0x80000000) >> 31; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2318 | hx = (hx & 0x000FFFFF) | ((0x3FF - i) << 20); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2319 | y = k + i; [ 120s] | ~~~~~~~~~~ [ 120s] 2320 | SET_HIGH_WORD(x, hx); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2321 | SET_LOW_WORD(x, lx); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2322 | [ 120s] | [ 120s] 2323 | double z = y * log10_2lo + ivln10 * log(x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2324 | return z + y * log10_2hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2325 | } [ 120s] | ~ [ 120s] 2326 | [ 120s] | [ 120s] 2327 | /* expm1(x) [ 120s] | ~~~~~~~~~~~ [ 120s] 2328 | * Returns exp(x)-1, the exponential of x minus 1. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2329 | * [ 120s] | ~ [ 120s] 2330 | * Method [ 120s] | ~~~~~~~~ [ 120s] 2331 | * 1. Argument reduction: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2332 | * Given x, find r and integer k such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2333 | * [ 120s] | ~ [ 120s] 2334 | * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2335 | * [ 120s] | ~ [ 120s] 2336 | * Here a correction term c will be computed to compensate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2337 | * the error in r when rounded to a floating-point number. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2338 | * [ 120s] | ~ [ 120s] 2339 | * 2. Approximating expm1(r) by a special rational function on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2340 | * the interval [0,0.34658]: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2341 | * Since [ 120s] | ~~~~~~~~ [ 120s] 2342 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2343 | * we define R1(r*r) by [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2344 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2345 | * That is, [ 120s] | ~~~~~~~~~~~ [ 120s] 2346 | * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2347 | * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2348 | * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2349 | * We use a special Reme algorithm on [0,0.347] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2350 | * a polynomial of degree 5 in r*r to approximate R1. The [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2351 | * maximum error of this polynomial approximation is bounded [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2352 | * by 2**-61. In other words, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2353 | * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2354 | * where Q1 = -1.6666666666666567384E-2, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2355 | * Q2 = 3.9682539681370365873E-4, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2356 | * Q3 = -9.9206344733435987357E-6, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2357 | * Q4 = 2.5051361420808517002E-7, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2358 | * Q5 = -6.2843505682382617102E-9; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2359 | * z = r*r, [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2360 | * with error bounded by [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2361 | * | 5 | -61 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2362 | * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2363 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2364 | * [ 120s] | ~ [ 120s] 2365 | * expm1(r) = exp(r)-1 is then computed by the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2366 | * specific way which minimize the accumulation rounding error: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2367 | * 2 3 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2368 | * r r [ 3 - (R1 + R1*r/2) ] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2369 | * expm1(r) = r + --- + --- * [--------------------] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2370 | * 2 2 [ 6 - r*(3 - R1*r/2) ] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2371 | * [ 120s] | ~ [ 120s] 2372 | * To compensate the error in the argument reduction, we use [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2373 | * expm1(r+c) = expm1(r) + c + expm1(r)*c [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2374 | * ~ expm1(r) + c + r*c [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2375 | * Thus c+r*c will be added in as the correction terms for [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2376 | * expm1(r+c). Now rearrange the term to avoid optimization [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2377 | * screw up: [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2378 | * ( 2 2 ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2379 | * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2380 | * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2381 | * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2382 | * ( ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2383 | * [ 120s] | ~ [ 120s] 2384 | * = r - E [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2385 | * 3. Scale back to obtain expm1(x): [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2386 | * From step 1, we have [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2387 | * expm1(x) = either 2^k*[expm1(r)+1] - 1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2388 | * = or 2^k*[expm1(r) + (1-2^-k)] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2389 | * 4. Implementation notes: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2390 | * (A). To save one multiplication, we scale the coefficient Qi [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2391 | * to Qi*2^i, and replace z by (x^2)/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2392 | * (B). To achieve maximum accuracy, we compute expm1(x) by [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2393 | * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2394 | * (ii) if k=0, return r-E [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2395 | * (iii) if k=-1, return 0.5*(r-E)-0.5 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2396 | * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2397 | * else return 1.0+2.0*(r-E); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2398 | * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2399 | * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2400 | * (vii) return 2^k(1-((E+2^-k)-r)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2401 | * [ 120s] | ~ [ 120s] 2402 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2403 | * expm1(INF) is INF, expm1(NaN) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2404 | * expm1(-INF) is -1, and [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2405 | * for finite argument, only expm1(0)=0 is exact. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2406 | * [ 120s] | ~ [ 120s] 2407 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 2408 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2409 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2410 | * [ 120s] | ~ [ 120s] 2411 | * Misc. info. [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2412 | * For IEEE double [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2413 | * if x > 7.09782712893383973096e+02 then expm1(x) overflow [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2414 | * [ 120s] | ~ [ 120s] 2415 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 2416 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2417 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2418 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2419 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2420 | */ [ 120s] | ~~ [ 120s] 2421 | double expm1(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2422 | { [ 120s] | ~ [ 120s] 2423 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2424 | one [ 120s] | ~~~ [ 120s] 2425 | = 1.0, [ 120s] | ~~~~~~ [ 120s] 2426 | tiny = 1.0e-300, [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2427 | o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2428 | ln2_hi = 6.93147180369123816490e-01, /* 0x3FE62E42, 0xFEE00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2429 | ln2_lo = 1.90821492927058770002e-10, /* 0x3DEA39EF, 0x35793C76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2430 | invln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2431 | /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2432 | x*x/2: */ [ 120s] | ~~~~~~~~~ [ 120s] 2433 | Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2434 | Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2435 | Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2436 | Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2437 | Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2438 | [ 120s] | [ 120s] 2439 | constexpr volatile double huge = 1.0e+300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2440 | [ 120s] | [ 120s] 2441 | double y, hi, lo, c, t, e, hxs, hfx, r1, twopk; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2442 | int32_t k, xsb; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2443 | uint32_t hx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2444 | [ 120s] | [ 120s] 2445 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2446 | xsb = hx & 0x80000000; /* sign bit of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2447 | hx &= 0x7FFFFFFF; /* high word of |x| */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2448 | [ 120s] | [ 120s] 2449 | /* filter out huge and non-finite argument */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2450 | if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2451 | if (hx >= 0x40862E42) { /* if |x|>=709.78... */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2452 | if (hx >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2453 | uint32_t low; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2454 | GET_LOW_WORD(low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2455 | if (((hx & 0xFFFFF) | low) != 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2456 | return x + x; /* NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2457 | else [ 120s] | ~~~~ [ 120s] 2458 | return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2459 | } [ 120s] | ~ [ 120s] 2460 | if (x > o_threshold) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2461 | return huge * huge; /* overflow */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2462 | } [ 120s] | ~ [ 120s] 2463 | if (xsb != 0 && x + tiny < 0.0) { /* x < -56*ln2, return -1.0 with inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2464 | /* raise inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2465 | return tiny - one; /* return -1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2466 | } [ 120s] | ~ [ 120s] 2467 | } [ 120s] | ~ [ 120s] 2468 | [ 120s] | [ 120s] 2469 | /* argument reduction */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2470 | if (hx > 0x3FD62E42) { /* if |x| > 0.5 ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2471 | if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2472 | if (xsb == 0) { [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2473 | hi = x - ln2_hi; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2474 | lo = ln2_lo; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2475 | k = 1; [ 120s] | ~~~~~~ [ 120s] 2476 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2477 | hi = x + ln2_hi; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2478 | lo = -ln2_lo; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2479 | k = -1; [ 120s] | ~~~~~~~ [ 120s] 2480 | } [ 120s] | ~ [ 120s] 2481 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2482 | k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2483 | t = k; [ 120s] | ~~~~~~ [ 120s] 2484 | hi = x - t * ln2_hi; /* t*ln2_hi is exact here */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2485 | lo = t * ln2_lo; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2486 | } [ 120s] | ~ [ 120s] 2487 | STRICT_ASSIGN(double, x, hi - lo); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2488 | c = (hi - x) - lo; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2489 | } else if (hx < 0x3C900000) { /* when |x|<2**-54, return x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2490 | t = huge + x; /* return x with inexact flags when x!=0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2491 | return x - (t - (huge + x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2492 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2493 | k = 0; [ 120s] | ~~~~~~ [ 120s] 2494 | } [ 120s] | ~ [ 120s] 2495 | [ 120s] | [ 120s] 2496 | /* x is now in primary range */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2497 | hfx = 0.5 * x; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2498 | hxs = x * hfx; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2499 | r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5)))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2500 | t = 3.0 - r1 * hfx; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2501 | e = hxs * ((r1 - t) / (6.0 - x * t)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2502 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2503 | return x - (x * e - hxs); /* c is 0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2504 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2505 | INSERT_WORDS(twopk, 0x3FF00000 + (k << 20), 0); /* 2^k */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2506 | e = (x * (e - c) - c); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2507 | e -= hxs; [ 120s] | ~~~~~~~~~ [ 120s] 2508 | if (k == -1) [ 120s] | ~~~~~~~~~~~~ [ 120s] 2509 | return 0.5 * (x - e) - 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2510 | if (k == 1) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2511 | if (x < -0.25) [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2512 | return -2.0 * (e - (x + 0.5)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2513 | else [ 120s] | ~~~~ [ 120s] 2514 | return one + 2.0 * (x - e); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2515 | } [ 120s] | ~ [ 120s] 2516 | if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2517 | y = one - (e - x); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2518 | // TODO(mvstanton): is this replacement for the hex float [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2519 | // sufficient? [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2520 | // if (k == 1024) y = y*2.0*0x1p1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2521 | if (k == 1024) [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2522 | y = y * 2.0 * 8.98846567431158e+307; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2523 | else [ 120s] | ~~~~ [ 120s] 2524 | y = y * twopk; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2525 | return y - one; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2526 | } [ 120s] | ~ [ 120s] 2527 | t = one; [ 120s] | ~~~~~~~~ [ 120s] 2528 | if (k < 20) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2529 | SET_HIGH_WORD(t, 0x3FF00000 - (0x200000 >> k)); /* t=1-2^-k */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2530 | y = t - (e - x); [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2531 | y = y * twopk; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2532 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2533 | SET_HIGH_WORD(t, ((0x3FF - k) << 20)); /* 2^-k */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2534 | y = x - (e + t); [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2535 | y += one; [ 120s] | ~~~~~~~~~ [ 120s] 2536 | y = y * twopk; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2537 | } [ 120s] | ~ [ 120s] 2538 | } [ 120s] | ~ [ 120s] 2539 | return y; [ 120s] | ~~~~~~~~~ [ 120s] 2540 | } [ 120s] | ~ [ 120s] 2541 | [ 120s] | [ 120s] 2542 | double cbrt(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2543 | { [ 120s] | ~ [ 120s] 2544 | constexpr uint32_t [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2545 | B1 [ 120s] | ~~ [ 120s] 2546 | = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2547 | B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2548 | [ 120s] | [ 120s] 2549 | /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2550 | constexpr double P0 = 1.87595182427177009643, /* 0x3FFE03E6, 0x0F61E692 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2551 | P1 = -1.88497979543377169875, /* 0xBFFE28E0, 0x92F02420 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2552 | P2 = 1.621429720105354466140, /* 0x3FF9F160, 0x4A49D6C2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2553 | P3 = -0.758397934778766047437, /* 0xBFE844CB, 0xBEE751D9 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2554 | P4 = 0.145996192886612446982; /* 0x3FC2B000, 0xD4E4EDD7 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2555 | [ 120s] | [ 120s] 2556 | int32_t hx; [ 120s] | ~~~~~~~~~~~ [ 120s] 2557 | union { [ 120s] | ~~~~~~~ [ 120s] 2558 | double value; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2559 | uint64_t bits; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2560 | } u; [ 120s] | ~~~~ [ 120s] 2561 | double r, s, t = 0.0, w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2562 | uint32_t sign; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2563 | uint32_t high, low; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2564 | [ 120s] | [ 120s] 2565 | EXTRACT_WORDS(hx, low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2566 | sign = hx & 0x80000000; /* sign= sign(x) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2567 | hx ^= sign; [ 120s] | ~~~~~~~~~~~ [ 120s] 2568 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2569 | return (x + x); /* cbrt(NaN,INF) is itself */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2570 | [ 120s] | [ 120s] 2571 | /* [ 120s] | ~~ [ 120s] 2572 | * Rough cbrt to 5 bits: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2573 | * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2574 | * where e is integral and >= 0, m is real and in [0, 1), and "/" and [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2575 | * "%" are integer division and modulus with rounding towards minus [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2576 | * infinity. The RHS is always >= the LHS and has a maximum relative [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2577 | * error of about 1 in 16. Adding a bias of -0.03306235651 to the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2578 | * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2579 | * floating point representation, for finite positive normal values, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2580 | * ordinary integer division of the value in bits magically gives [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2581 | * almost exactly the RHS of the above provided we first subtract the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2582 | * exponent bias (1023 for doubles) and later add it back. We do the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2583 | * subtraction virtually to keep e >= 0 so that ordinary integer [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2584 | * division rounds towards minus infinity; this is also efficient. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2585 | */ [ 120s] | ~~ [ 120s] 2586 | if (hx < 0x00100000) { /* zero or subnormal? */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2587 | if ((hx | low) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2588 | return (x); /* cbrt(0) is itself */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2589 | SET_HIGH_WORD(t, 0x43500000); /* set t= 2**54 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2590 | t *= x; [ 120s] | ~~~~~~~ [ 120s] 2591 | GET_HIGH_WORD(high, t); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2592 | INSERT_WORDS(t, sign | ((high & 0x7FFFFFFF) / 3 + B2), 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2593 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2594 | INSERT_WORDS(t, sign | (hx / 3 + B1), 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2595 | } [ 120s] | ~ [ 120s] 2596 | [ 120s] | [ 120s] 2597 | /* [ 120s] | ~~ [ 120s] 2598 | * New cbrt to 23 bits: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2599 | * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2600 | * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2601 | * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2602 | * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2603 | * gives us bounds for r = t**3/x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2604 | * [ 120s] | ~ [ 120s] 2605 | * Try to optimize for parallel evaluation as in k_tanf.c. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2606 | */ [ 120s] | ~~ [ 120s] 2607 | r = (t * t) * (t / x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2608 | t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2609 | [ 120s] | [ 120s] 2610 | /* [ 120s] | ~~ [ 120s] 2611 | * Round t away from zero to 23 bits (sloppily except for ensuring that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2612 | * the result is larger in magnitude than cbrt(x) but not much more than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2613 | * 2 23-bit ulps larger). With rounding towards zero, the error bound [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2614 | * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2615 | * in the rounded t, the infinite-precision error in the Newton [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2616 | * approximation barely affects third digit in the final error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2617 | * 0.667; the error in the rounded t can be up to about 3 23-bit ulps [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2618 | * before the final error is larger than 0.667 ulps. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2619 | */ [ 120s] | ~~ [ 120s] 2620 | u.value = t; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2621 | u.bits = (u.bits + 0x80000000) & 0xFFFFFFFFC0000000ULL; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2622 | t = u.value; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2623 | [ 120s] | [ 120s] 2624 | /* one step Newton iteration to 53 bits with error < 0.667 ulps */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2625 | s = t * t; /* t*t is exact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2626 | r = x / s; /* error <= 0.5 ulps; |r| < |t| */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2627 | w = t + t; /* t+t is exact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2628 | r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2629 | t = t + t * r; /* error <= 0.5 + 0.5/3 + epsilon */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2630 | [ 120s] | [ 120s] 2631 | return (t); [ 120s] | ~~~~~~~~~~~ [ 120s] 2632 | } [ 120s] | ~ [ 120s] 2633 | [ 120s] | [ 120s] 2634 | /* sin(x) [ 120s] | ~~~~~~~~~ [ 120s] 2635 | * Return sine function of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2636 | * [ 120s] | ~ [ 120s] 2637 | * kernel function: [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2638 | * __kernel_sin ... sine function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2639 | * __kernel_cos ... cose function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2640 | * __ieee754_rem_pio2 ... argument reduction routine [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2641 | * [ 120s] | ~ [ 120s] 2642 | * Method. [ 120s] | ~~~~~~~~~ [ 120s] 2643 | * Let S,C and T denote the sin, cos and tan respectively on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2644 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2645 | * in [-pi/4 , +pi/4], and let n = k mod 4. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2646 | * We have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2647 | * [ 120s] | ~ [ 120s] 2648 | * n sin(x) cos(x) tan(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2649 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2650 | * 0 S C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2651 | * 1 C -S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2652 | * 2 -S -C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2653 | * 3 -C S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2654 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2655 | * [ 120s] | ~ [ 120s] 2656 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2657 | * Let trig be any of sin, cos, or tan. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2658 | * trig(+-INF) is NaN, with signals; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2659 | * trig(NaN) is that NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2660 | * [ 120s] | ~ [ 120s] 2661 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 2662 | * TRIG(x) returns trig(x) nearly rounded [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2663 | */ [ 120s] | ~~ [ 120s] 2664 | double sin(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2665 | { [ 120s] | ~ [ 120s] 2666 | double y[2], z = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2667 | int32_t n, ix; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2668 | [ 120s] | [ 120s] 2669 | /* High word of x. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2670 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2671 | [ 120s] | [ 120s] 2672 | /* |x| ~< pi/4 */ [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2673 | ix &= 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2674 | if (ix <= 0x3FE921FB) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2675 | return __kernel_sin(x, z, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2676 | } else if (ix >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2677 | /* sin(Inf or NaN) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2678 | return x - x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2679 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2680 | /* argument reduction needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2681 | n = __ieee754_rem_pio2(x, y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2682 | switch (n & 3) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2683 | case 0: [ 120s] | ~~~~~~~ [ 120s] 2684 | return __kernel_sin(y[0], y[1], 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2685 | case 1: [ 120s] | ~~~~~~~ [ 120s] 2686 | return __kernel_cos(y[0], y[1]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2687 | case 2: [ 120s] | ~~~~~~~ [ 120s] 2688 | return -__kernel_sin(y[0], y[1], 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2689 | default: [ 120s] | ~~~~~~~~ [ 120s] 2690 | return -__kernel_cos(y[0], y[1]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2691 | } [ 120s] | ~ [ 120s] 2692 | } [ 120s] | ~ [ 120s] 2693 | } [ 120s] | ~ [ 120s] 2694 | [ 120s] | [ 120s] 2695 | /* tan(x) [ 120s] | ~~~~~~~~~ [ 120s] 2696 | * Return tangent function of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2697 | * [ 120s] | ~ [ 120s] 2698 | * kernel function: [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2699 | * __kernel_tan ... tangent function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2700 | * __ieee754_rem_pio2 ... argument reduction routine [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2701 | * [ 120s] | ~ [ 120s] 2702 | * Method. [ 120s] | ~~~~~~~~~ [ 120s] 2703 | * Let S,C and T denote the sin, cos and tan respectively on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2704 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2705 | * in [-pi/4 , +pi/4], and let n = k mod 4. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2706 | * We have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2707 | * [ 120s] | ~ [ 120s] 2708 | * n sin(x) cos(x) tan(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2709 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2710 | * 0 S C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2711 | * 1 C -S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2712 | * 2 -S -C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2713 | * 3 -C S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2714 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2715 | * [ 120s] | ~ [ 120s] 2716 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2717 | * Let trig be any of sin, cos, or tan. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2718 | * trig(+-INF) is NaN, with signals; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2719 | * trig(NaN) is that NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2720 | * [ 120s] | ~ [ 120s] 2721 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 2722 | * TRIG(x) returns trig(x) nearly rounded [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2723 | */ [ 120s] | ~~ [ 120s] 2724 | double tan(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2725 | { [ 120s] | ~ [ 120s] 2726 | double y[2], z = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2727 | int32_t n, ix; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2728 | [ 120s] | [ 120s] 2729 | /* High word of x. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2730 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2731 | [ 120s] | [ 120s] 2732 | /* |x| ~< pi/4 */ [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2733 | ix &= 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2734 | if (ix <= 0x3FE921FB) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2735 | return __kernel_tan(x, z, 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2736 | } else if (ix >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2737 | /* tan(Inf or NaN) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2738 | return x - x; /* NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2739 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2740 | /* argument reduction needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2741 | n = __ieee754_rem_pio2(x, y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2742 | /* 1 -> n even, -1 -> n odd */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2743 | return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2744 | } [ 120s] | ~ [ 120s] 2745 | } [ 120s] | ~ [ 120s] 2746 | [ 120s] | [ 120s] 2747 | /* [ 120s] | ~~ [ 120s] 2748 | * ES6 draft 09-27-13, section 20.2.2.12. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2749 | * Math.cosh [ 120s] | ~~~~~~~~~~~ [ 120s] 2750 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2751 | * mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2752 | * 1. Replace x by |x| (cosh(x) = cosh(-x)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2753 | * 2. [ 120s] | ~~~~~~~~~ [ 120s] 2754 | * [ exp(x) - 1 ]^2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2755 | * 0 <= x <= ln2/2 : cosh(x) := 1 + ------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2756 | * 2*exp(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2757 | * [ 120s] | ~ [ 120s] 2758 | * exp(x) + 1/exp(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2759 | * ln2/2 <= x <= 22 : cosh(x) := ------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2760 | * 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2761 | * 22 <= x <= lnovft : cosh(x) := exp(x)/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2762 | * lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2763 | * ln2ovft < x : cosh(x) := huge*huge (overflow) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2764 | * [ 120s] | ~ [ 120s] 2765 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2766 | * cosh(x) is |x| if x is +INF, -INF, or NaN. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2767 | * only cosh(0)=1 is exact for finite x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2768 | */ [ 120s] | ~~ [ 120s] 2769 | double cosh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2770 | { [ 120s] | ~ [ 120s] 2771 | constexpr double KCOSH_OVERFLOW = 710.4758600739439; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2772 | constexpr double one = 1.0, half = 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2773 | constexpr volatile double huge = 1.0e+300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2774 | [ 120s] | [ 120s] 2775 | int32_t ix; [ 120s] | ~~~~~~~~~~~ [ 120s] 2776 | [ 120s] | [ 120s] 2777 | /* High word of |x|. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2778 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2779 | ix &= 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2780 | [ 120s] | [ 120s] 2781 | // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2782 | if (ix < 0x3FD62E43) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2783 | double t = expm1(fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2784 | double w = one + t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2785 | // For |x| < 2^-55, cosh(x) = 1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2786 | if (ix < 0x3C800000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2787 | return w; [ 120s] | ~~~~~~~~~ [ 120s] 2788 | return one + (t * t) / (w + w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2789 | } [ 120s] | ~ [ 120s] 2790 | [ 120s] | [ 120s] 2791 | // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2792 | if (ix < 0x40360000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2793 | double t = exp(fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2794 | return half * t + half / t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2795 | } [ 120s] | ~ [ 120s] 2796 | [ 120s] | [ 120s] 2797 | // |x| in [22, log(maxdouble)], return half*exp(|x|) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2798 | if (ix < 0x40862E42) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2799 | return half * exp(fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2800 | [ 120s] | [ 120s] 2801 | // |x| in [log(maxdouble), overflowthreshold] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2802 | if (fabs(x) <= KCOSH_OVERFLOW) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2803 | double w = exp(half * fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2804 | double t = half * w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2805 | return t * w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2806 | } [ 120s] | ~ [ 120s] 2807 | [ 120s] | [ 120s] 2808 | /* x is INF or NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2809 | if (ix >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2810 | return x * x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2811 | [ 120s] | [ 120s] 2812 | // |x| > overflowthreshold. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2813 | return huge * huge; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2814 | } [ 120s] | ~ [ 120s] 2815 | [ 120s] | [ 120s] 2816 | /* [ 120s] | ~~ [ 120s] 2817 | * ES6 draft 09-27-13, section 20.2.2.30. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2818 | * Math.sinh [ 120s] | ~~~~~~~~~~~ [ 120s] 2819 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2820 | * mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2821 | * 1. Replace x by |x| (sinh(-x) = -sinh(x)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2822 | * 2. [ 120s] | ~~~~~~~~~ [ 120s] 2823 | * E + E/(E+1) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2824 | * 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2825 | * 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2826 | * [ 120s] | ~ [ 120s] 2827 | * 22 <= x <= lnovft : sinh(x) := exp(x)/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2828 | * lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2829 | * ln2ovft < x : sinh(x) := x*shuge (overflow) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2830 | * [ 120s] | ~ [ 120s] 2831 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2832 | * sinh(x) is |x| if x is +Infinity, -Infinity, or NaN. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2833 | * only sinh(0)=0 is exact for finite x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2834 | */ [ 120s] | ~~ [ 120s] 2835 | double sinh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2836 | { [ 120s] | ~ [ 120s] 2837 | constexpr double KSINH_OVERFLOW = 710.4758600739439, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2838 | TWO_M28 = 3.725290298461914e-9, // 2^-28, empty lower half [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2839 | LOG_MAXD = 709.7822265625; // 0x40862E42 00000000, empty lower half [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2840 | constexpr double shuge = 1.0e307; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2841 | [ 120s] | [ 120s] 2842 | double h = (x < 0) ? -0.5 : 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2843 | // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2844 | double ax = fabs(x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2845 | if (ax < 22) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2846 | // For |x| < 2^-28, sinh(x) = x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2847 | if (ax < TWO_M28) [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2848 | return x; [ 120s] | ~~~~~~~~~ [ 120s] 2849 | double t = expm1(ax); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2850 | if (ax < 1) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2851 | return h * (2 * t - t * t / (t + 1)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2852 | } [ 120s] | ~ [ 120s] 2853 | return h * (t + t / (t + 1)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2854 | } [ 120s] | ~ [ 120s] 2855 | // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2856 | if (ax < LOG_MAXD) [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2857 | return h * exp(ax); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2858 | // |x| in [log(maxdouble), overflowthreshold] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2859 | // overflowthreshold = 710.4758600739426 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2860 | if (ax <= KSINH_OVERFLOW) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2861 | double w = exp(0.5 * ax); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2862 | double t = h * w; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2863 | return t * w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2864 | } [ 120s] | ~ [ 120s] 2865 | // |x| > overflowthreshold or is NaN. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2866 | // Return Infinity of the appropriate sign or NaN. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2867 | return x * shuge; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2868 | } [ 120s] | ~ [ 120s] 2869 | [ 120s] | [ 120s] 2870 | /* Tanh(x) [ 120s] | ~~~~~~~~~~ [ 120s] 2871 | * Return the Hyperbolic Tangent of x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2872 | * [ 120s] | ~ [ 120s] 2873 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2874 | * x -x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2875 | * e - e [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2876 | * 0. tanh(x) is defined to be ----------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2877 | * x -x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2878 | * e + e [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2879 | * 1. reduce x to non-negative by tanh(-x) = -tanh(x). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2880 | * 2. 0 <= x < 2**-28 : tanh(x) := x with inexact if x != 0 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2881 | * -t [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2882 | * 2**-28 <= x < 1 : tanh(x) := -----; t = expm1(-2x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2883 | * t + 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2884 | * 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2885 | * 1 <= x < 22 : tanh(x) := 1 - -----; t = expm1(2x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2886 | * t + 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2887 | * 22 <= x <= INF : tanh(x) := 1. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2888 | * [ 120s] | ~ [ 120s] 2889 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2890 | * tanh(NaN) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2891 | * only tanh(0)=0 is exact for finite argument. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2892 | */ [ 120s] | ~~ [ 120s] 2893 | double tanh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2894 | { [ 120s] | ~ [ 120s] 2895 | constexpr volatile double tiny = 1.0e-300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2896 | constexpr double one = 1.0, two = 2.0, huge = 1.0e300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2897 | double t, z; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2898 | int32_t jx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2899 | [ 120s] | [ 120s] 2900 | GET_HIGH_WORD(jx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2901 | ix = jx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2902 | [ 120s] | [ 120s] 2903 | /* x is INF or NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2904 | if (ix >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2905 | if (jx >= 0) [ 120s] | ~~~~~~~~~~~~ [ 120s] 2906 | return one / x + one; /* tanh(+-inf)=+-1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2907 | else [ 120s] | ~~~~ [ 120s] 2908 | return one / x - one; /* tanh(NaN) = NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2909 | } [ 120s] | ~ [ 120s] 2910 | [ 120s] | [ 120s] 2911 | /* |x| < 22 */ [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2912 | if (ix < 0x40360000) { /* |x|<22 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2913 | if (ix < 0x3E300000 && huge + x > one) { /* |x|<2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2914 | return x; /* tanh(tiny) = tiny with inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2915 | } [ 120s] | ~ [ 120s] 2916 | if (ix >= 0x3FF00000) { /* |x|>=1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2917 | t = expm1(two * fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2918 | z = one - two / (t + two); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2919 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2920 | t = expm1(-two * fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2921 | z = -t / (t + two); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2922 | } [ 120s] | ~ [ 120s] 2923 | /* |x| >= 22, return +-1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2924 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2925 | z = one - tiny; /* raise inexact flag */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2926 | } [ 120s] | ~ [ 120s] 2927 | return (jx >= 0) ? z : -z; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2928 | } [ 120s] | ~ [ 120s] 2929 | [ 120s] | [ 120s] 2930 | double ceil(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2931 | { [ 120s] | ~ [ 120s] 2932 | constexpr double huge = 1.0e300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2933 | int i0, i1, j0; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2934 | unsigned i, j; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2935 | i0 = __HI(x); [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2936 | i1 = __LO(x); [ 120s] | ~~~~~~~~~~~ [ 120s] /home/abuild/rpmbuild/BUILD/lwnode-1.0.0/lwnode/code/escargotshim/deps/escargot/src/runtime/IEEE754.cpp:2936:10: note: in expansion of macro '__LO' [ 120s] 2936 | i1 = __LO(x); [ 120s] | ^~~~ [ 120s] /home/abuild/rpmbuild/BUILD/lwnode-1.0.0/lwnode/code/escargotshim/deps/escargot/src/runtime/IEEE754.cpp:86:18: warning: dereferencing type-punned pointer will break strict-aliasing rules [-Wstrict-aliasing] [ 120s] 86 | #define __LO(x) *(int *)&x [ 120s] | ^~~~~~~~~ [ 120s] 87 | #define __HIp(x) *(1 + (int *)x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 88 | #define __LOp(x) *(int *)x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 89 | [ 120s] | [ 120s] 90 | #else [ 120s] | ~~~~~ [ 120s] 91 | [ 120s] | [ 120s] 92 | typedef union { [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 93 | double value; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 94 | struct { [ 120s] | ~~~~~~~~ [ 120s] 95 | uint32_t msw; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 96 | uint32_t lsw; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 97 | } parts; [ 120s] | ~~~~~~~~ [ 120s] 98 | struct { [ 120s] | ~~~~~~~~ [ 120s] 99 | uint64_t w; [ 120s] | ~~~~~~~~~~~ [ 120s] 100 | } xparts; [ 120s] | ~~~~~~~~~ [ 120s] 101 | } ieee_double_shape_type; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 102 | [ 120s] | [ 120s] 103 | #define __HI(x) *(int *)&x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 104 | #define __LO(x) *(1 + (int *)&x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 105 | #define __HIp(x) *(int *)x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 106 | #define __LOp(x) *(1 + (int *)x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 107 | [ 120s] | [ 120s] 108 | #endif [ 120s] | ~~~~~~ [ 120s] 109 | [ 120s] | [ 120s] 110 | /* Get two 32 bit ints from a double. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 111 | [ 120s] | [ 120s] 112 | #define EXTRACT_WORDS(ix0, ix1, d) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 113 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 114 | ieee_double_shape_type ew_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 115 | ew_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 116 | (ix0) = ew_u.parts.msw; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 117 | (ix1) = ew_u.parts.lsw; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 118 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 119 | [ 120s] | [ 120s] 120 | /* Get a 64-bit int from a double. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 121 | #define EXTRACT_WORD64(ix, d) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 122 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 123 | ieee_double_shape_type ew_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 124 | ew_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 125 | (ix) = ew_u.xparts.w; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 126 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 127 | [ 120s] | [ 120s] 128 | /* Get the more significant 32 bit int from a double. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 129 | [ 120s] | [ 120s] 130 | #define GET_HIGH_WORD(i, d) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 131 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 132 | ieee_double_shape_type gh_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 133 | gh_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 134 | (i) = gh_u.parts.msw; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 135 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 136 | [ 120s] | [ 120s] 137 | /* Get the less significant 32 bit int from a double. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 138 | [ 120s] | [ 120s] 139 | #define GET_LOW_WORD(i, d) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 140 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 141 | ieee_double_shape_type gl_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 142 | gl_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 143 | (i) = gl_u.parts.lsw; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 144 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 145 | [ 120s] | [ 120s] 146 | /* Set a double from two 32 bit ints. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 147 | [ 120s] | [ 120s] 148 | #define INSERT_WORDS(d, ix0, ix1) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 149 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 150 | ieee_double_shape_type iw_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 151 | iw_u.parts.msw = (ix0); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 152 | iw_u.parts.lsw = (ix1); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 153 | (d) = iw_u.value; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 154 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 155 | [ 120s] | [ 120s] 156 | /* Set a double from a 64-bit int. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 157 | #define INSERT_WORD64(d, ix) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 158 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 159 | ieee_double_shape_type iw_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 160 | iw_u.xparts.w = (ix); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 161 | (d) = iw_u.value; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 162 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 163 | [ 120s] | [ 120s] 164 | /* Set the more significant 32 bits of a double from an int. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 165 | [ 120s] | [ 120s] 166 | #define SET_HIGH_WORD(d, v) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 167 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 168 | ieee_double_shape_type sh_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 169 | sh_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 170 | sh_u.parts.msw = (v); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 171 | (d) = sh_u.value; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 172 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 173 | [ 120s] | [ 120s] 174 | /* Set the less significant 32 bits of a double from an int. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 175 | [ 120s] | [ 120s] 176 | #define SET_LOW_WORD(d, v) \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 177 | do { \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 178 | ieee_double_shape_type sl_u; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 179 | sl_u.value = (d); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 180 | sl_u.parts.lsw = (v); \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 181 | (d) = sl_u.value; \ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 182 | } while (0) [ 120s] | ~~~~~~~~~~~ [ 120s] 183 | [ 120s] | [ 120s] 184 | /* Support macro. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 185 | [ 120s] | [ 120s] 186 | #define STRICT_ASSIGN(type, lval, rval) ((lval) = (rval)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 187 | [ 120s] | [ 120s] 188 | int32_t __ieee754_rem_pio2(double x, double *y) WARN_UNUSED_RETURN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 189 | double __kernel_cos(double x, double y) WARN_UNUSED_RETURN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 190 | int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 191 | const int32_t *ipio2) WARN_UNUSED_RETURN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 192 | double __kernel_sin(double x, double y, int iy) WARN_UNUSED_RETURN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 193 | [ 120s] | [ 120s] 194 | /* __ieee754_rem_pio2(x,y) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 195 | * [ 120s] | ~ [ 120s] 196 | * return the remainder of x rem pi/2 in y[0]+y[1] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 197 | * use __kernel_rem_pio2() [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 198 | */ [ 120s] | ~~ [ 120s] 199 | int32_t __ieee754_rem_pio2(double x, double *y) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 200 | { [ 120s] | ~ [ 120s] 201 | /* [ 120s] | ~~ [ 120s] 202 | * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 203 | */ [ 120s] | ~~ [ 120s] 204 | constexpr int32_t two_over_pi[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 205 | 0xA2F983, [ 120s] | ~~~~~~~~~ [ 120s] 206 | 0x6E4E44, [ 120s] | ~~~~~~~~~ [ 120s] 207 | 0x1529FC, [ 120s] | ~~~~~~~~~ [ 120s] 208 | 0x2757D1, [ 120s] | ~~~~~~~~~ [ 120s] 209 | 0xF534DD, [ 120s] | ~~~~~~~~~ [ 120s] 210 | 0xC0DB62, [ 120s] | ~~~~~~~~~ [ 120s] 211 | 0x95993C, [ 120s] | ~~~~~~~~~ [ 120s] 212 | 0x439041, [ 120s] | ~~~~~~~~~ [ 120s] 213 | 0xFE5163, [ 120s] | ~~~~~~~~~ [ 120s] 214 | 0xABDEBB, [ 120s] | ~~~~~~~~~ [ 120s] 215 | 0xC561B7, [ 120s] | ~~~~~~~~~ [ 120s] 216 | 0x246E3A, [ 120s] | ~~~~~~~~~ [ 120s] 217 | 0x424DD2, [ 120s] | ~~~~~~~~~ [ 120s] 218 | 0xE00649, [ 120s] | ~~~~~~~~~ [ 120s] 219 | 0x2EEA09, [ 120s] | ~~~~~~~~~ [ 120s] 220 | 0xD1921C, [ 120s] | ~~~~~~~~~ [ 120s] 221 | 0xFE1DEB, [ 120s] | ~~~~~~~~~ [ 120s] 222 | 0x1CB129, [ 120s] | ~~~~~~~~~ [ 120s] 223 | 0xA73EE8, [ 120s] | ~~~~~~~~~ [ 120s] 224 | 0x8235F5, [ 120s] | ~~~~~~~~~ [ 120s] 225 | 0x2EBB44, [ 120s] | ~~~~~~~~~ [ 120s] 226 | 0x84E99C, [ 120s] | ~~~~~~~~~ [ 120s] 227 | 0x7026B4, [ 120s] | ~~~~~~~~~ [ 120s] 228 | 0x5F7E41, [ 120s] | ~~~~~~~~~ [ 120s] 229 | 0x3991D6, [ 120s] | ~~~~~~~~~ [ 120s] 230 | 0x398353, [ 120s] | ~~~~~~~~~ [ 120s] 231 | 0x39F49C, [ 120s] | ~~~~~~~~~ [ 120s] 232 | 0x845F8B, [ 120s] | ~~~~~~~~~ [ 120s] 233 | 0xBDF928, [ 120s] | ~~~~~~~~~ [ 120s] 234 | 0x3B1FF8, [ 120s] | ~~~~~~~~~ [ 120s] 235 | 0x97FFDE, [ 120s] | ~~~~~~~~~ [ 120s] 236 | 0x05980F, [ 120s] | ~~~~~~~~~ [ 120s] 237 | 0xEF2F11, [ 120s] | ~~~~~~~~~ [ 120s] 238 | 0x8B5A0A, [ 120s] | ~~~~~~~~~ [ 120s] 239 | 0x6D1F6D, [ 120s] | ~~~~~~~~~ [ 120s] 240 | 0x367ECF, [ 120s] | ~~~~~~~~~ [ 120s] 241 | 0x27CB09, [ 120s] | ~~~~~~~~~ [ 120s] 242 | 0xB74F46, [ 120s] | ~~~~~~~~~ [ 120s] 243 | 0x3F669E, [ 120s] | ~~~~~~~~~ [ 120s] 244 | 0x5FEA2D, [ 120s] | ~~~~~~~~~ [ 120s] 245 | 0x7527BA, [ 120s] | ~~~~~~~~~ [ 120s] 246 | 0xC7EBE5, [ 120s] | ~~~~~~~~~ [ 120s] 247 | 0xF17B3D, [ 120s] | ~~~~~~~~~ [ 120s] 248 | 0x0739F7, [ 120s] | ~~~~~~~~~ [ 120s] 249 | 0x8A5292, [ 120s] | ~~~~~~~~~ [ 120s] 250 | 0xEA6BFB, [ 120s] | ~~~~~~~~~ [ 120s] 251 | 0x5FB11F, [ 120s] | ~~~~~~~~~ [ 120s] 252 | 0x8D5D08, [ 120s] | ~~~~~~~~~ [ 120s] 253 | 0x560330, [ 120s] | ~~~~~~~~~ [ 120s] 254 | 0x46FC7B, [ 120s] | ~~~~~~~~~ [ 120s] 255 | 0x6BABF0, [ 120s] | ~~~~~~~~~ [ 120s] 256 | 0xCFBC20, [ 120s] | ~~~~~~~~~ [ 120s] 257 | 0x9AF436, [ 120s] | ~~~~~~~~~ [ 120s] 258 | 0x1DA9E3, [ 120s] | ~~~~~~~~~ [ 120s] 259 | 0x91615E, [ 120s] | ~~~~~~~~~ [ 120s] 260 | 0xE61B08, [ 120s] | ~~~~~~~~~ [ 120s] 261 | 0x659985, [ 120s] | ~~~~~~~~~ [ 120s] 262 | 0x5F14A0, [ 120s] | ~~~~~~~~~ [ 120s] 263 | 0x68408D, [ 120s] | ~~~~~~~~~ [ 120s] 264 | 0xFFD880, [ 120s] | ~~~~~~~~~ [ 120s] 265 | 0x4D7327, [ 120s] | ~~~~~~~~~ [ 120s] 266 | 0x310606, [ 120s] | ~~~~~~~~~ [ 120s] 267 | 0x1556CA, [ 120s] | ~~~~~~~~~ [ 120s] 268 | 0x73A8C9, [ 120s] | ~~~~~~~~~ [ 120s] 269 | 0x60E27B, [ 120s] | ~~~~~~~~~ [ 120s] 270 | 0xC08C6B, [ 120s] | ~~~~~~~~~ [ 120s] 271 | }; [ 120s] | ~~ [ 120s] 272 | [ 120s] | [ 120s] 273 | constexpr int32_t npio2_hw[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 274 | 0x3FF921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 275 | 0x400921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 276 | 0x4012D97C, [ 120s] | ~~~~~~~~~~~ [ 120s] 277 | 0x401921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 278 | 0x401F6A7A, [ 120s] | ~~~~~~~~~~~ [ 120s] 279 | 0x4022D97C, [ 120s] | ~~~~~~~~~~~ [ 120s] 280 | 0x4025FDBB, [ 120s] | ~~~~~~~~~~~ [ 120s] 281 | 0x402921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 282 | 0x402C463A, [ 120s] | ~~~~~~~~~~~ [ 120s] 283 | 0x402F6A7A, [ 120s] | ~~~~~~~~~~~ [ 120s] 284 | 0x4031475C, [ 120s] | ~~~~~~~~~~~ [ 120s] 285 | 0x4032D97C, [ 120s] | ~~~~~~~~~~~ [ 120s] 286 | 0x40346B9C, [ 120s] | ~~~~~~~~~~~ [ 120s] 287 | 0x4035FDBB, [ 120s] | ~~~~~~~~~~~ [ 120s] 288 | 0x40378FDB, [ 120s] | ~~~~~~~~~~~ [ 120s] 289 | 0x403921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 290 | 0x403AB41B, [ 120s] | ~~~~~~~~~~~ [ 120s] 291 | 0x403C463A, [ 120s] | ~~~~~~~~~~~ [ 120s] 292 | 0x403DD85A, [ 120s] | ~~~~~~~~~~~ [ 120s] 293 | 0x403F6A7A, [ 120s] | ~~~~~~~~~~~ [ 120s] 294 | 0x40407E4C, [ 120s] | ~~~~~~~~~~~ [ 120s] 295 | 0x4041475C, [ 120s] | ~~~~~~~~~~~ [ 120s] 296 | 0x4042106C, [ 120s] | ~~~~~~~~~~~ [ 120s] 297 | 0x4042D97C, [ 120s] | ~~~~~~~~~~~ [ 120s] 298 | 0x4043A28C, [ 120s] | ~~~~~~~~~~~ [ 120s] 299 | 0x40446B9C, [ 120s] | ~~~~~~~~~~~ [ 120s] 300 | 0x404534AC, [ 120s] | ~~~~~~~~~~~ [ 120s] 301 | 0x4045FDBB, [ 120s] | ~~~~~~~~~~~ [ 120s] 302 | 0x4046C6CB, [ 120s] | ~~~~~~~~~~~ [ 120s] 303 | 0x40478FDB, [ 120s] | ~~~~~~~~~~~ [ 120s] 304 | 0x404858EB, [ 120s] | ~~~~~~~~~~~ [ 120s] 305 | 0x404921FB, [ 120s] | ~~~~~~~~~~~ [ 120s] 306 | }; [ 120s] | ~~ [ 120s] 307 | [ 120s] | [ 120s] 308 | /* [ 120s] | ~~ [ 120s] 309 | * invpio2: 53 bits of 2/pi [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 310 | * pio2_1: first 33 bit of pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 311 | * pio2_1t: pi/2 - pio2_1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 312 | * pio2_2: second 33 bit of pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 313 | * pio2_2t: pi/2 - (pio2_1+pio2_2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 314 | * pio2_3: third 33 bit of pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 315 | * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 316 | */ [ 120s] | ~~ [ 120s] 317 | [ 120s] | [ 120s] 318 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 319 | zero [ 120s] | ~~~~ [ 120s] 320 | = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 321 | half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 322 | two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 323 | invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 324 | pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 325 | pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 326 | pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 327 | pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 328 | pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 329 | pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 330 | [ 120s] | [ 120s] 331 | double z, w, t, r, fn; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 332 | double tx[3]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 333 | int32_t e0, i, j, nx, n, ix, hx; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 334 | uint32_t low; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 335 | [ 120s] | [ 120s] 336 | z = 0; [ 120s] | ~~~~~~ [ 120s] 337 | GET_HIGH_WORD(hx, x); /* high word of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 338 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 339 | if (ix <= 0x3FE921FB) { /* |x| ~<= pi/4 , no need for reduction */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 340 | y[0] = x; [ 120s] | ~~~~~~~~~ [ 120s] 341 | y[1] = 0; [ 120s] | ~~~~~~~~~ [ 120s] 342 | return 0; [ 120s] | ~~~~~~~~~ [ 120s] 343 | } [ 120s] | ~ [ 120s] 344 | if (ix < 0x4002D97C) { /* |x| < 3pi/4, special case with n=+-1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 345 | if (hx > 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 346 | z = x - pio2_1; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 347 | if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 348 | y[0] = z - pio2_1t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 349 | y[1] = (z - y[0]) - pio2_1t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 350 | } else { /* near pi/2, use 33+33+53 bit pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 351 | z -= pio2_2; [ 120s] | ~~~~~~~~~~~~ [ 120s] 352 | y[0] = z - pio2_2t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 353 | y[1] = (z - y[0]) - pio2_2t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 354 | } [ 120s] | ~ [ 120s] 355 | return 1; [ 120s] | ~~~~~~~~~ [ 120s] 356 | } else { /* negative x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 357 | z = x + pio2_1; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 358 | if (ix != 0x3FF921FB) { /* 33+53 bit pi is good enough */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 359 | y[0] = z + pio2_1t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 360 | y[1] = (z - y[0]) + pio2_1t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 361 | } else { /* near pi/2, use 33+33+53 bit pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 362 | z += pio2_2; [ 120s] | ~~~~~~~~~~~~ [ 120s] 363 | y[0] = z + pio2_2t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 364 | y[1] = (z - y[0]) + pio2_2t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 365 | } [ 120s] | ~ [ 120s] 366 | return -1; [ 120s] | ~~~~~~~~~~ [ 120s] 367 | } [ 120s] | ~ [ 120s] 368 | } [ 120s] | ~ [ 120s] 369 | if (ix <= 0x413921FB) { /* |x| ~<= 2^19*(pi/2), medium size */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 370 | t = fabs(x); [ 120s] | ~~~~~~~~~~~~ [ 120s] 371 | n = static_cast(t * invpio2 + half); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 372 | fn = static_cast(n); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 373 | r = t - fn * pio2_1; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 374 | w = fn * pio2_1t; /* 1st round good to 85 bit */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 375 | if (n < 32 && ix != npio2_hw[n - 1]) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 376 | y[0] = r - w; /* quick check no cancellation */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 377 | } else { [ 120s] | ~~~~~~~~ [ 120s] 378 | uint32_t high; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 379 | j = ix >> 20; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 380 | y[0] = r - w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 381 | GET_HIGH_WORD(high, y[0]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 382 | i = j - ((high >> 20) & 0x7FF); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 383 | if (i > 16) { /* 2nd iteration needed, good to 118 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 384 | t = r; [ 120s] | ~~~~~~ [ 120s] 385 | w = fn * pio2_2; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 386 | r = t - w; [ 120s] | ~~~~~~~~~~ [ 120s] 387 | w = fn * pio2_2t - ((t - r) - w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 388 | y[0] = r - w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 389 | GET_HIGH_WORD(high, y[0]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 390 | i = j - ((high >> 20) & 0x7FF); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 391 | if (i > 49) { /* 3rd iteration need, 151 bits acc */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 392 | t = r; /* will cover all possible cases */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 393 | w = fn * pio2_3; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 394 | r = t - w; [ 120s] | ~~~~~~~~~~ [ 120s] 395 | w = fn * pio2_3t - ((t - r) - w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 396 | y[0] = r - w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 397 | } [ 120s] | ~ [ 120s] 398 | } [ 120s] | ~ [ 120s] 399 | } [ 120s] | ~ [ 120s] 400 | y[1] = (r - y[0]) - w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 401 | if (hx < 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 402 | y[0] = -y[0]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 403 | y[1] = -y[1]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 404 | return -n; [ 120s] | ~~~~~~~~~~ [ 120s] 405 | } else { [ 120s] | ~~~~~~~~ [ 120s] 406 | return n; [ 120s] | ~~~~~~~~~ [ 120s] 407 | } [ 120s] | ~ [ 120s] 408 | } [ 120s] | ~ [ 120s] 409 | /* [ 120s] | ~~ [ 120s] 410 | * all other (large) arguments [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 411 | */ [ 120s] | ~~ [ 120s] 412 | if (ix >= 0x7FF00000) { /* x is inf or NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 413 | y[0] = y[1] = x - x; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 414 | return 0; [ 120s] | ~~~~~~~~~ [ 120s] 415 | } [ 120s] | ~ [ 120s] 416 | /* set z = scalbn(|x|,ilogb(x)-23) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 417 | GET_LOW_WORD(low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 418 | SET_LOW_WORD(z, low); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 419 | e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 420 | SET_HIGH_WORD(z, ix - static_cast(e0 << 20)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 421 | for (i = 0; i < 2; i++) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 422 | tx[i] = static_cast(static_cast(z)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 423 | z = (z - tx[i]) * two24; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 424 | } [ 120s] | ~ [ 120s] 425 | tx[2] = z; [ 120s] | ~~~~~~~~~~ [ 120s] 426 | nx = 3; [ 120s] | ~~~~~~~ [ 120s] 427 | while (tx[nx - 1] == zero) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 428 | nx--; /* skip zero term */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 429 | n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 430 | if (hx < 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 431 | y[0] = -y[0]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 432 | y[1] = -y[1]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 433 | return -n; [ 120s] | ~~~~~~~~~~ [ 120s] 434 | } [ 120s] | ~ [ 120s] 435 | return n; [ 120s] | ~~~~~~~~~ [ 120s] 436 | } [ 120s] | ~ [ 120s] 437 | [ 120s] | [ 120s] 438 | /* __kernel_cos( x, y ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 439 | * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 440 | * Input x is assumed to be bounded by ~pi/4 in magnitude. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 441 | * Input y is the tail of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 442 | * [ 120s] | ~ [ 120s] 443 | * Algorithm [ 120s] | ~~~~~~~~~~~ [ 120s] 444 | * 1. Since cos(-x) = cos(x), we need only to consider positive x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 445 | * 2. if x < 2^-27 (hx<0x3E400000 0), return 1 with inexact if x!=0. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 446 | * 3. cos(x) is approximated by a polynomial of degree 14 on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 447 | * [0,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 448 | * 4 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 449 | * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 450 | * where the remez error is [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 451 | * [ 120s] | ~ [ 120s] 452 | * | 2 4 6 8 10 12 14 | -58 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 453 | * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 454 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 455 | * [ 120s] | ~ [ 120s] 456 | * 4 6 8 10 12 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 457 | * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 458 | * cos(x) = 1 - x*x/2 + r [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 459 | * since cos(x+y) ~ cos(x) - sin(x)*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 460 | * ~ cos(x) - x*y, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 461 | * a correction term is necessary in cos(x) and hence [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 462 | * cos(x+y) = 1 - (x*x/2 - (r - x*y)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 463 | * For better accuracy when x > 0.3, let qx = |x|/4 with [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 464 | * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 465 | * Then [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 466 | * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 467 | * Note that 1-qx and (x*x/2-qx) is EXACT here, and the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 468 | * magnitude of the latter is at least a quarter of x*x/2, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 469 | * thus, reducing the rounding error in the subtraction. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 470 | */ [ 120s] | ~~ [ 120s] 471 | ALWAYS_INLINE double __kernel_cos(double x, double y) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 472 | { [ 120s] | ~ [ 120s] 473 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 474 | one [ 120s] | ~~~ [ 120s] 475 | = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 476 | C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 477 | C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 478 | C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 479 | C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 480 | C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 481 | C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 482 | [ 120s] | [ 120s] 483 | double a, iz, z, r, qx; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 484 | int32_t ix; [ 120s] | ~~~~~~~~~~~ [ 120s] 485 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 486 | ix &= 0x7FFFFFFF; /* ix = |x|'s high word*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 487 | if (ix < 0x3E400000 && static_cast(x) == 0) { /* if x < 2**27 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 488 | return one; /* generate inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 489 | } [ 120s] | ~ [ 120s] 490 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 491 | r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 492 | if (ix < 0x3FD33333) { /* if |x| < 0.3 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 493 | return one - (0.5 * z - (z * r - x * y)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 494 | } else { [ 120s] | ~~~~~~~~ [ 120s] 495 | if (ix > 0x3FE90000) { /* x > 0.78125 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 496 | qx = 0.28125; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 497 | } else { [ 120s] | ~~~~~~~~ [ 120s] 498 | INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 499 | } [ 120s] | ~ [ 120s] 500 | iz = 0.5 * z - qx; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 501 | a = one - qx; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 502 | return a - (iz - (z * r - x * y)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 503 | } [ 120s] | ~ [ 120s] 504 | } [ 120s] | ~ [ 120s] 505 | [ 120s] | [ 120s] 506 | /* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 507 | * double x[],y[]; int e0,nx,prec; int ipio2[]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 508 | * [ 120s] | ~ [ 120s] 509 | * __kernel_rem_pio2 return the last three digits of N with [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 510 | * y = x - N*pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 511 | * so that |y| < pi/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 512 | * [ 120s] | ~ [ 120s] 513 | * The method is to compute the integer (mod 8) and fraction parts of [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 514 | * (2/pi)*x without doing the full multiplication. In general we [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 515 | * skip the part of the product that are known to be a huge integer ( [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 516 | * more accurately, = 0 mod 8 ). Thus the number of operations are [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 517 | * independent of the exponent of the input. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 518 | * [ 120s] | ~ [ 120s] 519 | * (2/pi) is represented by an array of 24-bit integers in ipio2[]. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 520 | * [ 120s] | ~ [ 120s] 521 | * Input parameters: [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 522 | * x[] The input value (must be positive) is broken into nx [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 523 | * pieces of 24-bit integers in double precision format. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 524 | * x[i] will be the i-th 24 bit of x. The scaled exponent [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 525 | * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 526 | * match x's up to 24 bits. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 527 | * [ 120s] | ~ [ 120s] 528 | * Example of breaking a double positive z into x[0]+x[1]+x[2]: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 529 | * e0 = ilogb(z)-23 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 530 | * z = scalbn(z,-e0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 531 | * for i = 0,1,2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 532 | * x[i] = floor(z) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 533 | * z = (z-x[i])*2**24 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 534 | * [ 120s] | ~ [ 120s] 535 | * [ 120s] | ~ [ 120s] 536 | * y[] output result in an array of double precision numbers. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 537 | * The dimension of y[] is: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 538 | * 24-bit precision 1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 539 | * 53-bit precision 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 540 | * 64-bit precision 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 541 | * 113-bit precision 3 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 542 | * The actual value is the sum of them. Thus for 113-bit [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 543 | * precison, one may have to do something like: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 544 | * [ 120s] | ~ [ 120s] 545 | * long double t,w,r_head, r_tail; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 546 | * t = (long double)y[2] + (long double)y[1]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 547 | * w = (long double)y[0]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 548 | * r_head = t+w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 549 | * r_tail = w - (r_head - t); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 550 | * [ 120s] | ~ [ 120s] 551 | * e0 The exponent of x[0] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 552 | * [ 120s] | ~ [ 120s] 553 | * nx dimension of x[] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 554 | * [ 120s] | ~ [ 120s] 555 | * prec an integer indicating the precision: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 556 | * 0 24 bits (single) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 557 | * 1 53 bits (double) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 558 | * 2 64 bits (extended) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 559 | * 3 113 bits (quad) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 560 | * [ 120s] | ~ [ 120s] 561 | * ipio2[] [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 562 | * integer array, contains the (24*i)-th to (24*i+23)-th [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 563 | * bit of 2/pi after binary point. The corresponding [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 564 | * floating value is [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 565 | * [ 120s] | ~ [ 120s] 566 | * ipio2[i] * 2^(-24(i+1)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 567 | * [ 120s] | ~ [ 120s] 568 | * External function: [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 569 | * double scalbn(), floor(); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 570 | * [ 120s] | ~ [ 120s] 571 | * [ 120s] | ~ [ 120s] 572 | * Here is the description of some local variables: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 573 | * [ 120s] | ~ [ 120s] 574 | * jk jk+1 is the initial number of terms of ipio2[] needed [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 575 | * in the computation. The recommended value is 2,3,4, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 576 | * 6 for single, double, extended,and quad. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 577 | * [ 120s] | ~ [ 120s] 578 | * jz local integer variable indicating the number of [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 579 | * terms of ipio2[] used. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 580 | * [ 120s] | ~ [ 120s] 581 | * jx nx - 1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 582 | * [ 120s] | ~ [ 120s] 583 | * jv index for pointing to the suitable ipio2[] for the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 584 | * computation. In general, we want [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 585 | * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 586 | * is an integer. Thus [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 587 | * e0-3-24*jv >= 0 or (e0-3)/24 >= jv [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 588 | * Hence jv = max(0,(e0-3)/24). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 589 | * [ 120s] | ~ [ 120s] 590 | * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 591 | * [ 120s] | ~ [ 120s] 592 | * q[] double array with integral value, representing the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 593 | * 24-bits chunk of the product of x and 2/pi. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 594 | * [ 120s] | ~ [ 120s] 595 | * q0 the corresponding exponent of q[0]. Note that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 596 | * exponent for q[i] would be q0-24*i. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 597 | * [ 120s] | ~ [ 120s] 598 | * PIo2[] double precision array, obtained by cutting pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 599 | * into 24 bits chunks. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 600 | * [ 120s] | ~ [ 120s] 601 | * f[] ipio2[] in floating point [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 602 | * [ 120s] | ~ [ 120s] 603 | * iq[] integer array by breaking up q[] in 24-bits chunk. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 604 | * [ 120s] | ~ [ 120s] 605 | * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 606 | * [ 120s] | ~ [ 120s] 607 | * ih integer. If >0 it indicates q[] is >= 0.5, hence [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 608 | * it also indicates the *sign* of the result. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 609 | * [ 120s] | ~ [ 120s] 610 | */ [ 120s] | ~~ [ 120s] 611 | int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 612 | const int32_t *ipio2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 613 | { [ 120s] | ~ [ 120s] 614 | /* Constants: [ 120s] | ~~~~~~~~~~~~~ [ 120s] 615 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 616 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 617 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 618 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 619 | */ [ 120s] | ~~ [ 120s] 620 | constexpr int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 621 | [ 120s] | [ 120s] 622 | constexpr double PIo2[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 623 | 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 624 | 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 625 | 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 626 | 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 627 | 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 628 | 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 629 | 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 630 | 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 631 | }; [ 120s] | ~~ [ 120s] 632 | [ 120s] | [ 120s] 633 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 634 | zero [ 120s] | ~~~~ [ 120s] 635 | = 0.0, [ 120s] | ~~~~~~ [ 120s] 636 | one = 1.0, [ 120s] | ~~~~~~~~~~ [ 120s] 637 | two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 638 | twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 639 | [ 120s] | [ 120s] 640 | int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 641 | double z, fw, f[20], fq[20], q[20]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 642 | [ 120s] | [ 120s] 643 | /* initialize jk*/ [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 644 | jk = init_jk[prec]; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 645 | jp = jk; [ 120s] | ~~~~~~~~ [ 120s] 646 | [ 120s] | [ 120s] 647 | /* determine jx,jv,q0, note that 3>q0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 648 | jx = nx - 1; [ 120s] | ~~~~~~~~~~~~ [ 120s] 649 | jv = (e0 - 3) / 24; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 650 | if (jv < 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 651 | jv = 0; [ 120s] | ~~~~~~~ [ 120s] 652 | q0 = e0 - 24 * (jv + 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 653 | [ 120s] | [ 120s] 654 | /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 655 | j = jv - jx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 656 | m = jx + jk; [ 120s] | ~~~~~~~~~~~~ [ 120s] 657 | for (i = 0; i <= m; i++, j++) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 658 | f[i] = (j < 0) ? zero : static_cast(ipio2[j]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 659 | } [ 120s] | ~ [ 120s] 660 | [ 120s] | [ 120s] 661 | /* compute q[0],q[1],...q[jk] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 662 | for (i = 0; i <= jk; i++) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 663 | for (j = 0, fw = 0.0; j <= jx; j++) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 664 | fw += x[j] * f[jx + i - j]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 665 | q[i] = fw; [ 120s] | ~~~~~~~~~~ [ 120s] 666 | } [ 120s] | ~ [ 120s] 667 | [ 120s] | [ 120s] 668 | jz = jk; [ 120s] | ~~~~~~~~ [ 120s] 669 | recompute: [ 120s] | ~~~~~~~~~~ [ 120s] 670 | /* distill q[] into iq[] reversingly */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 671 | for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 672 | fw = static_cast(static_cast(twon24 * z)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 673 | iq[i] = static_cast(z - two24 * fw); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 674 | z = q[j - 1] + fw; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 675 | } [ 120s] | ~ [ 120s] 676 | [ 120s] | [ 120s] 677 | /* compute n */ [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 678 | z = scalbn(z, q0); /* actual value of z */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 679 | z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 680 | n = static_cast(z); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 681 | z -= static_cast(n); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 682 | ih = 0; [ 120s] | ~~~~~~~ [ 120s] 683 | if (q0 > 0) { /* need iq[jz-1] to determine n */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 684 | i = (iq[jz - 1] >> (24 - q0)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 685 | n += i; [ 120s] | ~~~~~~~ [ 120s] 686 | iq[jz - 1] -= i << (24 - q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 687 | ih = iq[jz - 1] >> (23 - q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 688 | } else if (q0 == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 689 | ih = iq[jz - 1] >> 23; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 690 | } else if (z >= 0.5) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 691 | ih = 2; [ 120s] | ~~~~~~~ [ 120s] 692 | } [ 120s] | ~ [ 120s] 693 | [ 120s] | [ 120s] 694 | if (ih > 0) { /* q > 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 695 | n += 1; [ 120s] | ~~~~~~~ [ 120s] 696 | carry = 0; [ 120s] | ~~~~~~~~~~ [ 120s] 697 | for (i = 0; i < jz; i++) { /* compute 1-q */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 698 | j = iq[i]; [ 120s] | ~~~~~~~~~~ [ 120s] 699 | if (carry == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 700 | if (j != 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 701 | carry = 1; [ 120s] | ~~~~~~~~~~ [ 120s] 702 | iq[i] = 0x1000000 - j; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 703 | } [ 120s] | ~ [ 120s] 704 | } else { [ 120s] | ~~~~~~~~ [ 120s] 705 | iq[i] = 0xFFFFFF - j; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 706 | } [ 120s] | ~ [ 120s] 707 | } [ 120s] | ~ [ 120s] 708 | if (q0 > 0) { /* rare case: chance is 1 in 12 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 709 | switch (q0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 710 | case 1: [ 120s] | ~~~~~~~ [ 120s] 711 | iq[jz - 1] &= 0x7FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 712 | break; [ 120s] | ~~~~~~ [ 120s] 713 | case 2: [ 120s] | ~~~~~~~ [ 120s] 714 | iq[jz - 1] &= 0x3FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 715 | break; [ 120s] | ~~~~~~ [ 120s] 716 | } [ 120s] | ~ [ 120s] 717 | } [ 120s] | ~ [ 120s] 718 | if (ih == 2) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 719 | z = one - z; [ 120s] | ~~~~~~~~~~~~ [ 120s] 720 | if (carry != 0) [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 721 | z -= scalbn(one, q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 722 | } [ 120s] | ~ [ 120s] 723 | } [ 120s] | ~ [ 120s] 724 | [ 120s] | [ 120s] 725 | /* check if recomputation is needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 726 | if (z == zero) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 727 | j = 0; [ 120s] | ~~~~~~ [ 120s] 728 | for (i = jz - 1; i >= jk; i--) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 729 | j |= iq[i]; [ 120s] | ~~~~~~~~~~~ [ 120s] 730 | if (j == 0) { /* need recomputation */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 731 | for (k = 1; jk >= k && iq[jk - k] == 0; k++) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 732 | /* k = no. of terms needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 733 | } [ 120s] | ~ [ 120s] 734 | [ 120s] | [ 120s] 735 | for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 736 | f[jx + i] = ipio2[jv + i]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 737 | for (j = 0, fw = 0.0; j <= jx; j++) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 738 | fw += x[j] * f[jx + i - j]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 739 | q[i] = fw; [ 120s] | ~~~~~~~~~~ [ 120s] 740 | } [ 120s] | ~ [ 120s] 741 | jz += k; [ 120s] | ~~~~~~~~ [ 120s] 742 | goto recompute; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 743 | } [ 120s] | ~ [ 120s] 744 | } [ 120s] | ~ [ 120s] 745 | [ 120s] | [ 120s] 746 | /* chop off zero terms */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 747 | if (z == 0.0) { [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 748 | jz -= 1; [ 120s] | ~~~~~~~~ [ 120s] 749 | q0 -= 24; [ 120s] | ~~~~~~~~~ [ 120s] 750 | while (iq[jz] == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 751 | jz--; [ 120s] | ~~~~~ [ 120s] 752 | q0 -= 24; [ 120s] | ~~~~~~~~~ [ 120s] 753 | } [ 120s] | ~ [ 120s] 754 | } else { /* break z into 24-bit if necessary */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 755 | z = scalbn(z, -q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 756 | if (z >= two24) { [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 757 | fw = static_cast(static_cast(twon24 * z)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 758 | iq[jz] = z - two24 * fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 759 | jz += 1; [ 120s] | ~~~~~~~~ [ 120s] 760 | q0 += 24; [ 120s] | ~~~~~~~~~ [ 120s] 761 | iq[jz] = fw; [ 120s] | ~~~~~~~~~~~~ [ 120s] 762 | } else { [ 120s] | ~~~~~~~~ [ 120s] 763 | iq[jz] = z; [ 120s] | ~~~~~~~~~~~ [ 120s] 764 | } [ 120s] | ~ [ 120s] 765 | } [ 120s] | ~ [ 120s] 766 | [ 120s] | [ 120s] 767 | /* convert integer "bit" chunk to floating-point value */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 768 | fw = scalbn(one, q0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 769 | for (i = jz; i >= 0; i--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 770 | q[i] = fw * iq[i]; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 771 | fw *= twon24; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 772 | } [ 120s] | ~ [ 120s] 773 | [ 120s] | [ 120s] 774 | /* compute PIo2[0,...,jp]*q[jz,...,0] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 775 | for (i = jz; i >= 0; i--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 776 | for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 777 | fw += PIo2[k] * q[i + k]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 778 | fq[jz - i] = fw; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 779 | } [ 120s] | ~ [ 120s] 780 | [ 120s] | [ 120s] 781 | /* compress fq[] into y[] */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 782 | switch (prec) { [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 783 | case 0: [ 120s] | ~~~~~~~ [ 120s] 784 | fw = 0.0; [ 120s] | ~~~~~~~~~ [ 120s] 785 | for (i = jz; i >= 0; i--) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 786 | fw += fq[i]; [ 120s] | ~~~~~~~~~~~~ [ 120s] 787 | y[0] = (ih == 0) ? fw : -fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 788 | break; [ 120s] | ~~~~~~ [ 120s] 789 | case 1: [ 120s] | ~~~~~~~ [ 120s] 790 | case 2: [ 120s] | ~~~~~~~ [ 120s] 791 | fw = 0.0; [ 120s] | ~~~~~~~~~ [ 120s] 792 | for (i = jz; i >= 0; i--) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 793 | fw += fq[i]; [ 120s] | ~~~~~~~~~~~~ [ 120s] 794 | y[0] = (ih == 0) ? fw : -fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 795 | fw = fq[0] - fw; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 796 | for (i = 1; i <= jz; i++) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 797 | fw += fq[i]; [ 120s] | ~~~~~~~~~~~~ [ 120s] 798 | y[1] = (ih == 0) ? fw : -fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 799 | break; [ 120s] | ~~~~~~ [ 120s] 800 | case 3: /* painful */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 801 | for (i = jz; i > 0; i--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 802 | fw = fq[i - 1] + fq[i]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 803 | fq[i] += fq[i - 1] - fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 804 | fq[i - 1] = fw; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 805 | } [ 120s] | ~ [ 120s] 806 | for (i = jz; i > 1; i--) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 807 | fw = fq[i - 1] + fq[i]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 808 | fq[i] += fq[i - 1] - fw; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 809 | fq[i - 1] = fw; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 810 | } [ 120s] | ~ [ 120s] 811 | for (fw = 0.0, i = jz; i >= 2; i--) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 812 | fw += fq[i]; [ 120s] | ~~~~~~~~~~~~ [ 120s] 813 | if (ih == 0) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 814 | y[0] = fq[0]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 815 | y[1] = fq[1]; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 816 | y[2] = fw; [ 120s] | ~~~~~~~~~~ [ 120s] 817 | } else { [ 120s] | ~~~~~~~~ [ 120s] 818 | y[0] = -fq[0]; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 819 | y[1] = -fq[1]; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 820 | y[2] = -fw; [ 120s] | ~~~~~~~~~~~ [ 120s] 821 | } [ 120s] | ~ [ 120s] 822 | } [ 120s] | ~ [ 120s] 823 | return n & 7; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 824 | } [ 120s] | ~ [ 120s] 825 | [ 120s] | [ 120s] 826 | /* __kernel_sin( x, y, iy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 827 | * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 828 | * Input x is assumed to be bounded by ~pi/4 in magnitude. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 829 | * Input y is the tail of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 830 | * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 831 | * [ 120s] | ~ [ 120s] 832 | * Algorithm [ 120s] | ~~~~~~~~~~~ [ 120s] 833 | * 1. Since sin(-x) = -sin(x), we need only to consider positive x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 834 | * 2. if x < 2^-27 (hx<0x3E400000 0), return x with inexact if x!=0. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 835 | * 3. sin(x) is approximated by a polynomial of degree 13 on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 836 | * [0,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 837 | * 3 13 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 838 | * sin(x) ~ x + S1*x + ... + S6*x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 839 | * where [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 840 | * [ 120s] | ~ [ 120s] 841 | * |sin(x) 2 4 6 8 10 12 | -58 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 842 | * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 843 | * | x | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 844 | * [ 120s] | ~ [ 120s] 845 | * 4. sin(x+y) = sin(x) + sin'(x')*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 846 | * ~ sin(x) + (1-x*x/2)*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 847 | * For better accuracy, let [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 848 | * 3 2 2 2 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 849 | * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 850 | * then 3 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 851 | * sin(x) = x + (S1*x + (x *(r-y/2)+y)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 852 | */ [ 120s] | ~~ [ 120s] 853 | ALWAYS_INLINE double __kernel_sin(double x, double y, int iy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 854 | { [ 120s] | ~ [ 120s] 855 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 856 | half [ 120s] | ~~~~ [ 120s] 857 | = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 858 | S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 859 | S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 860 | S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 861 | S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 862 | S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 863 | S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 864 | [ 120s] | [ 120s] 865 | double z, r, v; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 866 | int32_t ix; [ 120s] | ~~~~~~~~~~~ [ 120s] 867 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 868 | ix &= 0x7FFFFFFF; /* high word of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 869 | if (ix < 0x3E400000 && static_cast(x) == 0) { /* |x| < 2**-27 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 870 | return x; [ 120s] | ~~~~~~~~~ [ 120s] 871 | } /* generate inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 872 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 873 | v = z * x; [ 120s] | ~~~~~~~~~~ [ 120s] 874 | r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 875 | if (iy == 0) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 876 | return x + v * (S1 + z * r); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 877 | } else { [ 120s] | ~~~~~~~~ [ 120s] 878 | return x - ((z * (half * y - v * r) - y) - v * S1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 879 | } [ 120s] | ~ [ 120s] 880 | } [ 120s] | ~ [ 120s] 881 | [ 120s] | [ 120s] 882 | /* __kernel_tan( x, y, k ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 883 | * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 884 | * Input x is assumed to be bounded by ~pi/4 in magnitude. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 885 | * Input y is the tail of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 886 | * Input k indicates whether tan (if k=1) or [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 887 | * -1/tan (if k= -1) is returned. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 888 | * [ 120s] | ~ [ 120s] 889 | * Algorithm [ 120s] | ~~~~~~~~~~~ [ 120s] 890 | * 1. Since tan(-x) = -tan(x), we need only to consider positive x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 891 | * 2. if x < 2^-28 (hx<0x3E300000 0), return x with inexact if x!=0. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 892 | * 3. tan(x) is approximated by a odd polynomial of degree 27 on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 893 | * [0,0.67434] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 894 | * 3 27 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 895 | * tan(x) ~ x + T1*x + ... + T13*x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 896 | * where [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 897 | * [ 120s] | ~ [ 120s] 898 | * |tan(x) 2 4 26 | -59.2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 899 | * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 900 | * | x | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 901 | * [ 120s] | ~ [ 120s] 902 | * Note: tan(x+y) = tan(x) + tan'(x)*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 903 | * ~ tan(x) + (1+x*x)*y [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 904 | * Therefore, for better accuracy in computing tan(x+y), let [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 905 | * 3 2 2 2 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 906 | * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 907 | * then [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 908 | * 3 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 909 | * tan(x+y) = x + (T1*x + (x *(r+y)+y)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 910 | * [ 120s] | ~ [ 120s] 911 | * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 912 | * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 913 | * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 914 | */ [ 120s] | ~~ [ 120s] 915 | double __kernel_tan(double x, double y, int iy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 916 | { [ 120s] | ~ [ 120s] 917 | constexpr double xxx[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 918 | 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 919 | 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 920 | 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 921 | 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 922 | 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 923 | 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 924 | 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 925 | 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 926 | 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 927 | 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 928 | 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 929 | -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 930 | 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 931 | /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 932 | /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 933 | /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 934 | }; [ 120s] | ~~ [ 120s] 935 | #define one xxx[13] [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 936 | #define pio4 xxx[14] [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 937 | #define pio4lo xxx[15] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 938 | #define T xxx [ 120s] | ~~~~~~~~~~~~~ [ 120s] 939 | [ 120s] | [ 120s] 940 | double z, r, v, w, s; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 941 | int32_t ix, hx; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 942 | [ 120s] | [ 120s] 943 | GET_HIGH_WORD(hx, x); /* high word of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 944 | ix = hx & 0x7FFFFFFF; /* high word of |x| */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 945 | if (ix < 0x3E300000 && static_cast(x) == 0) { /* x < 2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 946 | /* generate inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 947 | uint32_t low; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 948 | GET_LOW_WORD(low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 949 | if (((ix | low) | (iy + 1)) == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 950 | return one / fabs(x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 951 | } else { [ 120s] | ~~~~~~~~ [ 120s] 952 | if (iy == 1) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 953 | return x; [ 120s] | ~~~~~~~~~ [ 120s] 954 | } else { /* compute -1 / (x+y) carefully */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 955 | double a, t; [ 120s] | ~~~~~~~~~~~~ [ 120s] 956 | [ 120s] | [ 120s] 957 | z = w = x + y; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 958 | SET_LOW_WORD(z, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 959 | v = y - (z - x); [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 960 | t = a = -one / w; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 961 | SET_LOW_WORD(t, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 962 | s = one + t * z; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 963 | return t + a * (s + t * v); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 964 | } [ 120s] | ~ [ 120s] 965 | } [ 120s] | ~ [ 120s] 966 | } [ 120s] | ~ [ 120s] 967 | if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 968 | if (hx < 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 969 | x = -x; [ 120s] | ~~~~~~~ [ 120s] 970 | y = -y; [ 120s] | ~~~~~~~ [ 120s] 971 | } [ 120s] | ~ [ 120s] 972 | z = pio4 - x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 973 | w = pio4lo - y; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 974 | x = z + w; [ 120s] | ~~~~~~~~~~ [ 120s] 975 | y = 0.0; [ 120s] | ~~~~~~~~ [ 120s] 976 | } [ 120s] | ~ [ 120s] 977 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 978 | w = z * z; [ 120s] | ~~~~~~~~~~ [ 120s] 979 | /* [ 120s] | ~~ [ 120s] 980 | * Break x^5*(T[1]+x^2*T[2]+...) into [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 981 | * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 982 | * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 983 | */ [ 120s] | ~~ [ 120s] 984 | r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11])))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 985 | v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12]))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 986 | s = z * x; [ 120s] | ~~~~~~~~~~ [ 120s] 987 | r = y + z * (s * (r + v) + y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 988 | r += T[0] * s; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 989 | w = x + r; [ 120s] | ~~~~~~~~~~ [ 120s] 990 | if (ix >= 0x3FE59428) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 991 | v = iy; [ 120s] | ~~~~~~~ [ 120s] 992 | return (1 - ((hx >> 30) & 2)) * (v - 2.0 * (x - (w * w / (w + v) - r))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 993 | } [ 120s] | ~ [ 120s] 994 | if (iy == 1) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 995 | return w; [ 120s] | ~~~~~~~~~ [ 120s] 996 | } else { [ 120s] | ~~~~~~~~ [ 120s] 997 | /* [ 120s] | ~~ [ 120s] 998 | * if allow error up to 2 ulp, simply return [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 999 | * -1.0 / (x+r) here [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1000 | */ [ 120s] | ~~ [ 120s] 1001 | /* compute -1.0 / (x+r) accurately */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1002 | double a, t; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1003 | z = w; [ 120s] | ~~~~~~ [ 120s] 1004 | SET_LOW_WORD(z, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1005 | v = r - (z - x); /* z+v = r+x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1006 | t = a = -1.0 / w; /* a = -1.0/w */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1007 | SET_LOW_WORD(t, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1008 | s = 1.0 + t * z; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1009 | return t + a * (s + t * v); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1010 | } [ 120s] | ~ [ 120s] 1011 | [ 120s] | [ 120s] 1012 | #undef one [ 120s] | ~~~~~~~~~~ [ 120s] 1013 | #undef pio4 [ 120s] | ~~~~~~~~~~~ [ 120s] 1014 | #undef pio4lo [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1015 | #undef T [ 120s] | ~~~~~~~~ [ 120s] 1016 | } [ 120s] | ~ [ 120s] 1017 | [ 120s] | [ 120s] 1018 | } // namespace [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1019 | [ 120s] | [ 120s] 1020 | /* acos(x) [ 120s] | ~~~~~~~~~~ [ 120s] 1021 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1022 | * acos(x) = pi/2 - asin(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1023 | * acos(-x) = pi/2 + asin(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1024 | * For |x|<=0.5 [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1025 | * acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1026 | * For x>0.5 [ 120s] | ~~~~~~~~~~~ [ 120s] 1027 | * acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1028 | * = 2asin(sqrt((1-x)/2)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1029 | * = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1030 | * = 2f + (2c + 2s*z*R(z)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1031 | * where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1032 | * for f so that f+c ~ sqrt(z). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1033 | * For x<-0.5 [ 120s] | ~~~~~~~~~~~~ [ 120s] 1034 | * acos(x) = pi - 2asin(sqrt((1-|x|)/2)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1035 | * = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1036 | * [ 120s] | ~ [ 120s] 1037 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1038 | * if x is NaN, return x itself; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1039 | * if |x|>1, return NaN with invalid signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1040 | * [ 120s] | ~ [ 120s] 1041 | * Function needed: sqrt [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1042 | */ [ 120s] | ~~ [ 120s] 1043 | double acos(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1044 | { [ 120s] | ~ [ 120s] 1045 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1046 | one [ 120s] | ~~~ [ 120s] 1047 | = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1048 | pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1049 | pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1050 | pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1051 | pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1052 | pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1053 | pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1054 | pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1055 | pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1056 | pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1057 | qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1058 | qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1059 | qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1060 | qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1061 | [ 120s] | [ 120s] 1062 | double z, p, q, r, w, s, c, df; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1063 | int32_t hx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1064 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1065 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1066 | if (ix >= 0x3FF00000) { /* |x| >= 1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1067 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1068 | GET_LOW_WORD(lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1069 | if (((ix - 0x3FF00000) | lx) == 0) { /* |x|==1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1070 | if (hx > 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1071 | return 0.0; /* acos(1) = 0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1072 | else [ 120s] | ~~~~ [ 120s] 1073 | return pi + 2.0 * pio2_lo; /* acos(-1)= pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1074 | } [ 120s] | ~ [ 120s] 1075 | return (x - x) / (x - x); /* acos(|x|>1) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1076 | } [ 120s] | ~ [ 120s] 1077 | if (ix < 0x3FE00000) { /* |x| < 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1078 | if (ix <= 0x3C600000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1079 | return pio2_hi + pio2_lo; /*if|x|<2**-57*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1080 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1081 | p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1082 | q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1083 | r = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1084 | return pio2_hi - (x - (pio2_lo - x * r)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1085 | } else if (hx < 0) { /* x < -0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1086 | z = (one + x) * 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1087 | p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1088 | q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1089 | s = sqrt(z); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1090 | r = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1091 | w = r * s - pio2_lo; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1092 | return pi - 2.0 * (s + w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1093 | } else { /* x > 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1094 | z = (one - x) * 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1095 | s = sqrt(z); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1096 | df = s; [ 120s] | ~~~~~~~ [ 120s] 1097 | SET_LOW_WORD(df, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1098 | c = (z - df * df) / (s + df); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1099 | p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1100 | q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1101 | r = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1102 | w = r * s + c; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1103 | return 2.0 * (df + w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1104 | } [ 120s] | ~ [ 120s] 1105 | } [ 120s] | ~ [ 120s] 1106 | [ 120s] | [ 120s] 1107 | /* acosh(x) [ 120s] | ~~~~~~~~~~~ [ 120s] 1108 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1109 | * Based on [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1110 | * acosh(x) = log [ x + sqrt(x*x-1) ] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1111 | * we have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1112 | * acosh(x) := log(x)+ln2, if x is large; else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1113 | * acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1114 | * acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1115 | * [ 120s] | ~ [ 120s] 1116 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1117 | * acosh(x) is NaN with signal if x<1. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1118 | * acosh(NaN) is NaN without signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1119 | */ [ 120s] | ~~ [ 120s] 1120 | double acosh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1121 | { [ 120s] | ~ [ 120s] 1122 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1123 | one [ 120s] | ~~~ [ 120s] 1124 | = 1.0, [ 120s] | ~~~~~~ [ 120s] 1125 | ln2 = 6.93147180559945286227e-01; /* 0x3FE62E42, 0xFEFA39EF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1126 | double t; [ 120s] | ~~~~~~~~~ [ 120s] 1127 | int32_t hx; [ 120s] | ~~~~~~~~~~~ [ 120s] 1128 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1129 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1130 | if (hx < 0x3FF00000) { /* x < 1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1131 | return (x - x) / (x - x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1132 | } else if (hx >= 0x41B00000) { /* x > 2**28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1133 | if (hx >= 0x7FF00000) { /* x is inf of NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1134 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1135 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1136 | return log(x) + ln2; /* acosh(huge)=log(2x) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1137 | } [ 120s] | ~ [ 120s] 1138 | } else if (((hx - 0x3FF00000) | lx) == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1139 | return 0.0; /* acosh(1) = 0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1140 | } else if (hx > 0x40000000) { /* 2**28 > x > 2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1141 | t = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1142 | return log(2.0 * x - one / (x + sqrt(t - one))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1143 | } else { /* 10.98 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1163 | * asin(x) = pi/2 - 2*(s+s*z*R(z)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1164 | * = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1165 | * For x<=0.98, let pio4_hi = pio2_hi/2, then [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1166 | * f = hi part of s; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1167 | * c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1168 | * and [ 120s] | ~~~~~~~~~~ [ 120s] 1169 | * asin(x) = pi/2 - 2*(s+s*z*R(z)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1170 | * = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1171 | * = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1172 | * [ 120s] | ~ [ 120s] 1173 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1174 | * if x is NaN, return x itself; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1175 | * if |x|>1, return NaN with invalid signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1176 | */ [ 120s] | ~~ [ 120s] 1177 | double asin(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1178 | { [ 120s] | ~ [ 120s] 1179 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1180 | one [ 120s] | ~~~ [ 120s] 1181 | = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1182 | huge = 1.000e+300, [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1183 | pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1184 | pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1185 | pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1186 | /* coefficient for R(x^2) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1187 | pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1188 | pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1189 | pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1190 | pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1191 | pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1192 | pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1193 | qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1194 | qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1195 | qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1196 | qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1197 | [ 120s] | [ 120s] 1198 | double t, w, p, q, c, r, s; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1199 | int32_t hx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1200 | [ 120s] | [ 120s] 1201 | t = 0; [ 120s] | ~~~~~~ [ 120s] 1202 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1203 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1204 | if (ix >= 0x3FF00000) { /* |x|>= 1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1205 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1206 | GET_LOW_WORD(lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1207 | if (((ix - 0x3FF00000) | lx) == 0) /* asin(1)=+-pi/2 with inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1208 | return x * pio2_hi + x * pio2_lo; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1209 | return (x - x) / (x - x); /* asin(|x|>1) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1210 | } else if (ix < 0x3FE00000) { /* |x|<0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1211 | if (ix < 0x3E400000 && huge + x > one) { /* if |x| < 2**-27 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1212 | return x; /* return x with inexact if x!=0*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1213 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1214 | t = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1215 | } [ 120s] | ~ [ 120s] 1216 | p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1217 | q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1218 | w = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1219 | return x + x * w; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1220 | } [ 120s] | ~ [ 120s] 1221 | /* 1> |x|>= 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1222 | w = one - fabs(x); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1223 | t = w * 0.5; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1224 | p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1225 | q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1226 | s = sqrt(t); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1227 | if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1228 | w = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1229 | t = pio2_hi - (2.0 * (s + s * w) - pio2_lo); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1230 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1231 | w = s; [ 120s] | ~~~~~~ [ 120s] 1232 | SET_LOW_WORD(w, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1233 | c = (t - w * w) / (s + w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1234 | r = p / q; [ 120s] | ~~~~~~~~~~ [ 120s] 1235 | p = 2.0 * s * r - (pio2_lo - 2.0 * c); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1236 | q = pio4_hi - 2.0 * w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1237 | t = pio4_hi - (p - q); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1238 | } [ 120s] | ~ [ 120s] 1239 | if (hx > 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1240 | return t; [ 120s] | ~~~~~~~~~ [ 120s] 1241 | else [ 120s] | ~~~~ [ 120s] 1242 | return -t; [ 120s] | ~~~~~~~~~~ [ 120s] 1243 | } [ 120s] | ~ [ 120s] 1244 | /* asinh(x) [ 120s] | ~~~~~~~~~~~ [ 120s] 1245 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1246 | * Based on [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1247 | * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1248 | * we have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1249 | * asinh(x) := x if 1+x*x=1, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1250 | * := sign(x)*(log(x)+ln2)) for large |x|, else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1251 | * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1252 | * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1253 | */ [ 120s] | ~~ [ 120s] 1254 | double asinh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1255 | { [ 120s] | ~ [ 120s] 1256 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1257 | one [ 120s] | ~~~ [ 120s] 1258 | = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1259 | ln2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1260 | huge = 1.00000000000000000000e+300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1261 | [ 120s] | [ 120s] 1262 | double t, w; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1263 | int32_t hx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1264 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1265 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1266 | if (ix >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1267 | return x + x; /* x is inf or NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1268 | if (ix < 0x3E300000 && huge + x > one) { /* |x|<2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1269 | return x; /* return x inexact except 0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1270 | } [ 120s] | ~ [ 120s] 1271 | if (ix > 0x41B00000) { /* |x| > 2**28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1272 | w = log(fabs(x)) + ln2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1273 | } else if (ix > 0x40000000) { /* 2**28 > |x| > 2.0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1274 | t = fabs(x); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1275 | w = log(2.0 * t + one / (sqrt(x * x + one) + t)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1276 | } else { /* 2.0 > |x| > 2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1277 | t = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1278 | w = log1p(fabs(x) + t / (one + sqrt(one + t))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1279 | } [ 120s] | ~ [ 120s] 1280 | if (hx > 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1281 | return w; [ 120s] | ~~~~~~~~~ [ 120s] 1282 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1283 | return -w; [ 120s] | ~~~~~~~~~~ [ 120s] 1284 | } [ 120s] | ~ [ 120s] 1285 | } [ 120s] | ~ [ 120s] 1286 | [ 120s] | [ 120s] 1287 | /* atan(x) [ 120s] | ~~~~~~~~~~ [ 120s] 1288 | * Method [ 120s] | ~~~~~~~~ [ 120s] 1289 | * 1. Reduce x to positive by atan(x) = -atan(-x). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1290 | * 2. According to the integer k=4t+0.25 chopped, t=x, the argument [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1291 | * is further reduced to one of the following intervals and the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1292 | * arctangent of t is evaluated by the corresponding formula: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1293 | * [ 120s] | ~ [ 120s] 1294 | * [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1295 | * [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1296 | * [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1297 | * [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1298 | * [39/16,INF] atan(x) = atan(INF) + atan( -1/t ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1299 | * [ 120s] | ~ [ 120s] 1300 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1301 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1302 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1303 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1304 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1305 | */ [ 120s] | ~~ [ 120s] 1306 | double atan(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1307 | { [ 120s] | ~ [ 120s] 1308 | constexpr double atanhi[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1309 | 4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1310 | 7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1311 | 9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1312 | 1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1313 | }; [ 120s] | ~~ [ 120s] 1314 | [ 120s] | [ 120s] 1315 | constexpr double atanlo[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1316 | 2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1317 | 3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1318 | 1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1319 | 6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1320 | }; [ 120s] | ~~ [ 120s] 1321 | [ 120s] | [ 120s] 1322 | constexpr double aT[] = { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1323 | 3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1324 | -1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1325 | 1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1326 | -1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1327 | 9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1328 | -7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1329 | 6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1330 | -5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1331 | 4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1332 | -3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1333 | 1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1334 | }; [ 120s] | ~~ [ 120s] 1335 | [ 120s] | [ 120s] 1336 | constexpr double one = 1.0, huge = 1.0e300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1337 | [ 120s] | [ 120s] 1338 | double w, s1, s2, z; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1339 | int32_t ix, hx, id; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1340 | [ 120s] | [ 120s] 1341 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1342 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1343 | if (ix >= 0x44100000) { /* if |x| >= 2^66 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1344 | uint32_t low; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1345 | GET_LOW_WORD(low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1346 | if (ix > 0x7FF00000 || (ix == 0x7FF00000 && (low != 0))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1347 | return x + x; /* NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1348 | if (hx > 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1349 | return atanhi[3] + *(volatile double *)&atanlo[3]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1350 | else [ 120s] | ~~~~ [ 120s] 1351 | return -atanhi[3] - *(volatile double *)&atanlo[3]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1352 | } [ 120s] | ~ [ 120s] 1353 | if (ix < 0x3FDC0000) { /* |x| < 0.4375 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1354 | if (ix < 0x3E400000 && huge + x > one) { /* |x| < 2^-27 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1355 | return x; /* raise inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1356 | } [ 120s] | ~ [ 120s] 1357 | id = -1; [ 120s] | ~~~~~~~~ [ 120s] 1358 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1359 | x = fabs(x); [ 120s] | ~~~~~~~~~~~~ [ 120s] 1360 | if (ix < 0x3FF30000) { /* |x| < 1.1875 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1361 | if (ix < 0x3FE60000) { /* 7/16 <=|x|<11/16 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1362 | id = 0; [ 120s] | ~~~~~~~ [ 120s] 1363 | x = (2.0 * x - one) / (2.0 + x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1364 | } else { /* 11/16<=|x|< 19/16 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1365 | id = 1; [ 120s] | ~~~~~~~ [ 120s] 1366 | x = (x - one) / (x + one); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1367 | } [ 120s] | ~ [ 120s] 1368 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1369 | if (ix < 0x40038000) { /* |x| < 2.4375 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1370 | id = 2; [ 120s] | ~~~~~~~ [ 120s] 1371 | x = (x - 1.5) / (one + 1.5 * x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1372 | } else { /* 2.4375 <= |x| < 2^66 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1373 | id = 3; [ 120s] | ~~~~~~~ [ 120s] 1374 | x = -1.0 / x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1375 | } [ 120s] | ~ [ 120s] 1376 | } [ 120s] | ~ [ 120s] 1377 | } [ 120s] | ~ [ 120s] 1378 | /* end of argument reduction */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1379 | z = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1380 | w = z * z; [ 120s] | ~~~~~~~~~~ [ 120s] 1381 | /* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1382 | s1 = z * (aT[0] + w * (aT[2] + w * (aT[4] + w * (aT[6] + w * (aT[8] + w * aT[10]))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1383 | s2 = w * (aT[1] + w * (aT[3] + w * (aT[5] + w * (aT[7] + w * aT[9])))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1384 | if (id < 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1385 | return x - x * (s1 + s2); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1386 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1387 | z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1388 | return (hx < 0) ? -z : z; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1389 | } [ 120s] | ~ [ 120s] 1390 | } [ 120s] | ~ [ 120s] 1391 | [ 120s] | [ 120s] 1392 | /* atan2(y,x) [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1393 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1394 | * 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1395 | * 2. Reduce x to positive by (if x and y are unexceptional): [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1396 | * ARG (x+iy) = arctan(y/x) ... if x > 0, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1397 | * ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1398 | * [ 120s] | ~ [ 120s] 1399 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1400 | * [ 120s] | ~ [ 120s] 1401 | * ATAN2((anything), NaN ) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1402 | * ATAN2(NAN , (anything) ) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1403 | * ATAN2(+-0, +(anything but NaN)) is +-0 ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1404 | * ATAN2(+-0, -(anything but NaN)) is +-pi ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1405 | * ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1406 | * ATAN2(+-(anything but INF and NaN), +INF) is +-0 ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1407 | * ATAN2(+-(anything but INF and NaN), -INF) is +-pi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1408 | * ATAN2(+-INF,+INF ) is +-pi/4 ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1409 | * ATAN2(+-INF,-INF ) is +-3pi/4; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1410 | * ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1411 | * [ 120s] | ~ [ 120s] 1412 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1413 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1414 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1415 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1416 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1417 | */ [ 120s] | ~~ [ 120s] 1418 | double atan2(double y, double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1419 | { [ 120s] | ~ [ 120s] 1420 | constexpr volatile double tiny = 1.0e-300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1421 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1422 | zero [ 120s] | ~~~~ [ 120s] 1423 | = 0.0, [ 120s] | ~~~~~~ [ 120s] 1424 | pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1425 | pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1426 | pi = 3.1415926535897931160E+00; /* 0x400921FB, 0x54442D18 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1427 | constexpr volatile double pi_lo = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1428 | [ 120s] | [ 120s] 1429 | double z; [ 120s] | ~~~~~~~~~ [ 120s] 1430 | int32_t k, m, hx, hy, ix, iy; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1431 | uint32_t lx, ly; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1432 | [ 120s] | [ 120s] 1433 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1434 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1435 | EXTRACT_WORDS(hy, ly, y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1436 | iy = hy & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1437 | if (((ix | ((lx | -static_cast(lx)) >> 31)) > 0x7FF00000) || ((iy | ((ly | -static_cast(ly)) >> 31)) > 0x7FF00000)) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1438 | return x + y; /* x or y is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1439 | } [ 120s] | ~ [ 120s] 1440 | if (((hx - 0x3FF00000) | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1441 | return atan(y); /* x=1.0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1442 | m = ((hy >> 31) & 1) | ((hx >> 30) & 2); /* 2*sign(x)+sign(y) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1443 | [ 120s] | [ 120s] 1444 | /* when y = 0 */ [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1445 | if ((iy | ly) == 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1446 | switch (m) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1447 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1448 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1449 | return y; /* atan(+-0,+anything)=+-0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1450 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1451 | return pi + tiny; /* atan(+0,-anything) = pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1452 | case 3: [ 120s] | ~~~~~~~ [ 120s] 1453 | return -pi - tiny; /* atan(-0,-anything) =-pi */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1454 | } [ 120s] | ~ [ 120s] 1455 | } [ 120s] | ~ [ 120s] 1456 | /* when x = 0 */ [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1457 | if ((ix | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1458 | return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1459 | [ 120s] | [ 120s] 1460 | /* when x is INF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1461 | if (ix == 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1462 | if (iy == 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1463 | switch (m) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1464 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1465 | return pi_o_4 + tiny; /* atan(+INF,+INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1466 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1467 | return -pi_o_4 - tiny; /* atan(-INF,+INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1468 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1469 | return 3.0 * pi_o_4 + tiny; /*atan(+INF,-INF)*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1470 | case 3: [ 120s] | ~~~~~~~ [ 120s] 1471 | return -3.0 * pi_o_4 - tiny; /*atan(-INF,-INF)*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1472 | } [ 120s] | ~ [ 120s] 1473 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1474 | switch (m) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1475 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1476 | return zero; /* atan(+...,+INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1477 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1478 | return -zero; /* atan(-...,+INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1479 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1480 | return pi + tiny; /* atan(+...,-INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1481 | case 3: [ 120s] | ~~~~~~~ [ 120s] 1482 | return -pi - tiny; /* atan(-...,-INF) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1483 | } [ 120s] | ~ [ 120s] 1484 | } [ 120s] | ~ [ 120s] 1485 | } [ 120s] | ~ [ 120s] 1486 | /* when y is INF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1487 | if (iy == 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1488 | return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1489 | [ 120s] | [ 120s] 1490 | /* compute y/x */ [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1491 | k = (iy - ix) >> 20; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1492 | if (k > 60) { /* |y/x| > 2**60 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1493 | z = pi_o_2 + 0.5 * pi_lo; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1494 | m &= 1; [ 120s] | ~~~~~~~ [ 120s] 1495 | } else if (hx < 0 && k < -60) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1496 | z = 0.0; /* 0 > |y|/x > -2**-60 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1497 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1498 | z = atan(fabs(y / x)); /* safe to do y/x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1499 | } [ 120s] | ~ [ 120s] 1500 | switch (m) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1501 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1502 | return z; /* atan(+,+) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1503 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1504 | return -z; /* atan(-,+) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1505 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1506 | return pi - (z - pi_lo); /* atan(+,-) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1507 | default: /* case 3 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1508 | return (z - pi_lo) - pi; /* atan(-,-) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1509 | } [ 120s] | ~ [ 120s] 1510 | } [ 120s] | ~ [ 120s] 1511 | [ 120s] | [ 120s] 1512 | /* cos(x) [ 120s] | ~~~~~~~~~ [ 120s] 1513 | * Return cosine function of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1514 | * [ 120s] | ~ [ 120s] 1515 | * kernel function: [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1516 | * __kernel_sin ... sine function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1517 | * __kernel_cos ... cosine function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1518 | * __ieee754_rem_pio2 ... argument reduction routine [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1519 | * [ 120s] | ~ [ 120s] 1520 | * Method. [ 120s] | ~~~~~~~~~ [ 120s] 1521 | * Let S,C and T denote the sin, cos and tan respectively on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1522 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1523 | * in [-pi/4 , +pi/4], and let n = k mod 4. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1524 | * We have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1525 | * [ 120s] | ~ [ 120s] 1526 | * n sin(x) cos(x) tan(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1527 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1528 | * 0 S C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1529 | * 1 C -S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1530 | * 2 -S -C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1531 | * 3 -C S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1532 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1533 | * [ 120s] | ~ [ 120s] 1534 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1535 | * Let trig be any of sin, cos, or tan. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1536 | * trig(+-INF) is NaN, with signals; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1537 | * trig(NaN) is that NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1538 | * [ 120s] | ~ [ 120s] 1539 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 1540 | * TRIG(x) returns trig(x) nearly rounded [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1541 | */ [ 120s] | ~~ [ 120s] 1542 | double cos(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1543 | { [ 120s] | ~ [ 120s] 1544 | double y[2], z = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1545 | int32_t n, ix; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1546 | [ 120s] | [ 120s] 1547 | /* High word of x. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1548 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1549 | [ 120s] | [ 120s] 1550 | /* |x| ~< pi/4 */ [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1551 | ix &= 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1552 | if (ix <= 0x3FE921FB) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1553 | return __kernel_cos(x, z); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1554 | } else if (ix >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1555 | /* cos(Inf or NaN) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1556 | return x - x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1557 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1558 | /* argument reduction needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1559 | n = __ieee754_rem_pio2(x, y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1560 | switch (n & 3) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1561 | case 0: [ 120s] | ~~~~~~~ [ 120s] 1562 | return __kernel_cos(y[0], y[1]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1563 | case 1: [ 120s] | ~~~~~~~ [ 120s] 1564 | return -__kernel_sin(y[0], y[1], 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1565 | case 2: [ 120s] | ~~~~~~~ [ 120s] 1566 | return -__kernel_cos(y[0], y[1]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1567 | default: [ 120s] | ~~~~~~~~ [ 120s] 1568 | return __kernel_sin(y[0], y[1], 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1569 | } [ 120s] | ~ [ 120s] 1570 | } [ 120s] | ~ [ 120s] 1571 | } [ 120s] | ~ [ 120s] 1572 | [ 120s] | [ 120s] 1573 | /* exp(x) [ 120s] | ~~~~~~~~~ [ 120s] 1574 | * Returns the exponential of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1575 | * [ 120s] | ~ [ 120s] 1576 | * Method [ 120s] | ~~~~~~~~ [ 120s] 1577 | * 1. Argument reduction: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1578 | * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1579 | * Given x, find r and integer k such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1580 | * [ 120s] | ~ [ 120s] 1581 | * x = k*ln2 + r, |r| <= 0.5*ln2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1582 | * [ 120s] | ~ [ 120s] 1583 | * Here r will be represented as r = hi-lo for better [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1584 | * accuracy. [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1585 | * [ 120s] | ~ [ 120s] 1586 | * 2. Approximation of exp(r) by a special rational function on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1587 | * the interval [0,0.34658]: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1588 | * Write [ 120s] | ~~~~~~~~~~~~ [ 120s] 1589 | * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ... [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1590 | * We use a special Remes algorithm on [0,0.34658] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1591 | * a polynomial of degree 5 to approximate R. The maximum error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1592 | * of this polynomial approximation is bounded by 2**-59. In [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1593 | * other words, [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1594 | * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1595 | * (where z=r*r, and the values of P1 to P5 are listed below) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1596 | * and [ 120s] | ~~~~~~~~~~ [ 120s] 1597 | * | 5 | -59 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1598 | * | 2.0+P1*z+...+P5*z - R(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1599 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1600 | * The computation of exp(r) thus becomes [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1601 | * 2*r [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1602 | * exp(r) = 1 + ------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1603 | * R - r [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1604 | * r*R1(r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1605 | * = 1 + r + ----------- (for better accuracy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1606 | * 2 - R1(r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1607 | * where [ 120s] | ~~~~~~~~~~~~ [ 120s] 1608 | * 2 4 10 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1609 | * R1(r) = r - (P1*r + P2*r + ... + P5*r ). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1610 | * [ 120s] | ~ [ 120s] 1611 | * 3. Scale back to obtain exp(x): [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1612 | * From step 1, we have [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1613 | * exp(x) = 2^k * exp(r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1614 | * [ 120s] | ~ [ 120s] 1615 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1616 | * exp(INF) is INF, exp(NaN) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1617 | * exp(-INF) is 0, and [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1618 | * for finite argument, only exp(0)=1 is exact. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1619 | * [ 120s] | ~ [ 120s] 1620 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 1621 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1622 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1623 | * [ 120s] | ~ [ 120s] 1624 | * Misc. info. [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1625 | * For IEEE double [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1626 | * if x > 7.09782712893383973096e+02 then exp(x) overflow [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1627 | * if x < -7.45133219101941108420e+02 then exp(x) underflow [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1628 | * [ 120s] | ~ [ 120s] 1629 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1630 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1631 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1632 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1633 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1634 | */ [ 120s] | ~~ [ 120s] 1635 | double exp(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1636 | { [ 120s] | ~ [ 120s] 1637 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1638 | one [ 120s] | ~~~ [ 120s] 1639 | = 1.0, [ 120s] | ~~~~~~ [ 120s] 1640 | halF[2] = { 0.5, -0.5 }, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1641 | o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1642 | u_threshold = -7.45133219101941108420e+02, /* 0xC0874910, 0xD52D3051 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1643 | ln2HI[2] = { 6.93147180369123816490e-01, /* 0x3FE62E42, 0xFEE00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1644 | -6.93147180369123816490e-01 }, /* 0xBFE62E42, 0xFEE00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1645 | ln2LO[2] = { 1.90821492927058770002e-10, /* 0x3DEA39EF, 0x35793C76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1646 | -1.90821492927058770002e-10 }, /* 0xBDEA39EF, 0x35793C76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1647 | invln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1648 | P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1649 | P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1650 | P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1651 | P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1652 | P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1653 | E = 2.718281828459045; /* 0x4005BF0A, 0x8B145769 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1654 | [ 120s] | [ 120s] 1655 | constexpr volatile double [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1656 | huge [ 120s] | ~~~~ [ 120s] 1657 | = 1.0e+300, [ 120s] | ~~~~~~~~~~~ [ 120s] 1658 | twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1659 | two1023 = 8.988465674311579539e307; /* 0x1p1023 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1660 | [ 120s] | [ 120s] 1661 | double y, hi = 0.0, lo = 0.0, c, t, twopk; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1662 | int32_t k = 0, xsb; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1663 | uint32_t hx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1664 | [ 120s] | [ 120s] 1665 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1666 | xsb = (hx >> 31) & 1; /* sign bit of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1667 | hx &= 0x7FFFFFFF; /* high word of |x| */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1668 | [ 120s] | [ 120s] 1669 | /* filter out non-finite argument */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1670 | if (hx >= 0x40862E42) { /* if |x|>=709.78... */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1671 | if (hx >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1672 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1673 | GET_LOW_WORD(lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1674 | if (((hx & 0xFFFFF) | lx) != 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1675 | return x + x; /* NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1676 | else [ 120s] | ~~~~ [ 120s] 1677 | return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1678 | } [ 120s] | ~ [ 120s] 1679 | if (x > o_threshold) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1680 | return huge * huge; /* overflow */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1681 | if (x < u_threshold) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1682 | return twom1000 * twom1000; /* underflow */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1683 | } [ 120s] | ~ [ 120s] 1684 | [ 120s] | [ 120s] 1685 | /* argument reduction */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1686 | if (hx > 0x3FD62E42) { /* if |x| > 0.5 ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1687 | if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1688 | /* TODO(rtoy): We special case exp(1) here to return the correct [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1689 | * value of E, as the computation below would get the last bit [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1690 | * wrong. We should probably fix the algorithm instead. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1691 | */ [ 120s] | ~~ [ 120s] 1692 | if (x == 1.0) [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1693 | return E; [ 120s] | ~~~~~~~~~ [ 120s] 1694 | hi = x - ln2HI[xsb]; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1695 | lo = ln2LO[xsb]; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1696 | k = 1 - xsb - xsb; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1697 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1698 | k = static_cast(invln2 * x + halF[xsb]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1699 | t = k; [ 120s] | ~~~~~~ [ 120s] 1700 | hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1701 | lo = t * ln2LO[0]; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1702 | } [ 120s] | ~ [ 120s] 1703 | STRICT_ASSIGN(double, x, hi - lo); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1704 | } else if (hx < 0x3E300000) { /* when |x|<2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1705 | if (huge + x > one) [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1706 | return one + x; /* trigger inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1707 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1708 | k = 0; [ 120s] | ~~~~~~ [ 120s] 1709 | } [ 120s] | ~ [ 120s] 1710 | [ 120s] | [ 120s] 1711 | /* x is now in primary range */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1712 | t = x * x; [ 120s] | ~~~~~~~~~~ [ 120s] 1713 | if (k >= -1021) { [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1714 | INSERT_WORDS(twopk, 0x3FF00000 + (k << 20), 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1715 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1716 | INSERT_WORDS(twopk, 0x3FF00000 + ((k + 1000) << 20), 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1717 | } [ 120s] | ~ [ 120s] 1718 | c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5)))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1719 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1720 | return one - ((x * c) / (c - 2.0) - x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1721 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1722 | y = one - ((lo - (x * c) / (2.0 - c)) - hi); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1723 | } [ 120s] | ~ [ 120s] 1724 | if (k >= -1021) { [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1725 | if (k == 1024) [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 1726 | return y * 2.0 * two1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1727 | return y * twopk; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1728 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1729 | return y * twopk * twom1000; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1730 | } [ 120s] | ~ [ 120s] 1731 | } [ 120s] | ~ [ 120s] 1732 | [ 120s] | [ 120s] 1733 | /* [ 120s] | ~~ [ 120s] 1734 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1735 | * 1.Reduced x to positive by atanh(-x) = -atanh(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1736 | * 2.For x>=0.5 [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1737 | * 1 2x x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1738 | * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1739 | * 2 1 - x 1 - x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1740 | * [ 120s] | ~ [ 120s] 1741 | * For x<0.5 [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1742 | * atanh(x) = 0.5*log1p(2x+2x*x/(1-x)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1743 | * [ 120s] | ~ [ 120s] 1744 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1745 | * atanh(x) is NaN if |x| > 1 with signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1746 | * atanh(NaN) is that NaN with no signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1747 | * atanh(+-1) is +-INF with signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1748 | * [ 120s] | ~ [ 120s] 1749 | */ [ 120s] | ~~ [ 120s] 1750 | double atanh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1751 | { [ 120s] | ~ [ 120s] 1752 | constexpr double one = 1.0, huge = 1e300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1753 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1754 | [ 120s] | [ 120s] 1755 | double t; [ 120s] | ~~~~~~~~~ [ 120s] 1756 | int32_t hx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1757 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1758 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1759 | ix = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1760 | if ((ix | ((lx | -static_cast(lx)) >> 31)) > 0x3FF00000) /* |x|>1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1761 | return (x - x) / (x - x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1762 | if (ix == 0x3FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1763 | return x / zero; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1764 | if (ix < 0x3E300000 && (huge + x) > zero) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1765 | return x; /* x<2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1766 | SET_HIGH_WORD(x, ix); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1767 | if (ix < 0x3FE00000) { /* x < 0.5 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1768 | t = x + x; [ 120s] | ~~~~~~~~~~ [ 120s] 1769 | t = 0.5 * log1p(t + t * x / (one - x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1770 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1771 | t = 0.5 * log1p((x + x) / (one - x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1772 | } [ 120s] | ~ [ 120s] 1773 | if (hx >= 0) [ 120s] | ~~~~~~~~~~~~ [ 120s] 1774 | return t; [ 120s] | ~~~~~~~~~ [ 120s] 1775 | else [ 120s] | ~~~~ [ 120s] 1776 | return -t; [ 120s] | ~~~~~~~~~~ [ 120s] 1777 | } [ 120s] | ~ [ 120s] 1778 | [ 120s] | [ 120s] 1779 | /* log(x) [ 120s] | ~~~~~~~~~ [ 120s] 1780 | * Return the logrithm of x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1781 | * [ 120s] | ~ [ 120s] 1782 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1783 | * 1. Argument Reduction: find k and f such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1784 | * x = 2^k * (1+f), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1785 | * where sqrt(2)/2 < 1+f < sqrt(2) . [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1786 | * [ 120s] | ~ [ 120s] 1787 | * 2. Approximation of log(1+f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1788 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1789 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1790 | * = 2s + s*R [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1791 | * We use a special Reme algorithm on [0,0.1716] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1792 | * a polynomial of degree 14 to approximate R The maximum error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1793 | * of this polynomial approximation is bounded by 2**-58.45. In [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1794 | * other words, [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1795 | * 2 4 6 8 10 12 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1796 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1797 | * (the values of Lg1 to Lg7 are listed in the program) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1798 | * and [ 120s] | ~~~~~~ [ 120s] 1799 | * | 2 14 | -58.45 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1800 | * | Lg1*s +...+Lg7*s - R(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1801 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1802 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1803 | * In order to guarantee error in log below 1ulp, we compute log [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1804 | * by [ 120s] | ~~~~~ [ 120s] 1805 | * log(1+f) = f - s*(f - R) (if f is not too large) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1806 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1807 | * [ 120s] | ~ [ 120s] 1808 | * 3. Finally, log(x) = k*ln2 + log(1+f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1809 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1810 | * Here ln2 is split into two floating point number: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1811 | * ln2_hi + ln2_lo, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1812 | * where n*ln2_hi is always exact for |n| < 2000. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1813 | * [ 120s] | ~ [ 120s] 1814 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1815 | * log(x) is NaN with signal if x < 0 (including -INF) ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1816 | * log(+INF) is +INF; log(0) is -INF with signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1817 | * log(NaN) is that NaN with no signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1818 | * [ 120s] | ~ [ 120s] 1819 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 1820 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1821 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1822 | * [ 120s] | ~ [ 120s] 1823 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1824 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1825 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1826 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1827 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1828 | */ [ 120s] | ~~ [ 120s] 1829 | double log(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1830 | { [ 120s] | ~ [ 120s] 1831 | constexpr double /* -- */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1832 | ln2_hi [ 120s] | ~~~~~~ [ 120s] 1833 | = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1834 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1835 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1836 | Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1837 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1838 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1839 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1840 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1841 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1842 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1843 | [ 120s] | [ 120s] 1844 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1845 | constexpr volatile double vzero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1846 | [ 120s] | [ 120s] 1847 | double hfsq, f, s, z, R, w, t1, t2, dk; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1848 | int32_t k, hx, i, j; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1849 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1850 | [ 120s] | [ 120s] 1851 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1852 | [ 120s] | [ 120s] 1853 | k = 0; [ 120s] | ~~~~~~ [ 120s] 1854 | if (hx < 0x00100000) { /* x < 2**-1022 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1855 | if (((hx & 0x7FFFFFFF) | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1856 | return -two54 / vzero; /* log(+-0)=-inf */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1857 | if (hx < 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1858 | return (x - x) / zero; /* log(-#) = NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1859 | k -= 54; [ 120s] | ~~~~~~~~ [ 120s] 1860 | x *= two54; /* subnormal number, scale up x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1861 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1862 | } [ 120s] | ~ [ 120s] 1863 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1864 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1865 | k += (hx >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1866 | hx &= 0x000FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1867 | i = (hx + 0x95F64) & 0x100000; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1868 | SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1869 | k += (i >> 20); [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1870 | f = x - 1.0; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1871 | if ((0x000FFFFF & (2 + hx)) < 3) { /* -2**-20 <= f < 2**-20 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1872 | if (f == zero) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1873 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1874 | return zero; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1875 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1876 | dk = static_cast(k); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1877 | return dk * ln2_hi + dk * ln2_lo; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1878 | } [ 120s] | ~ [ 120s] 1879 | } [ 120s] | ~ [ 120s] 1880 | R = f * f * (0.5 - 0.33333333333333333 * f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1881 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1882 | return f - R; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1883 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1884 | dk = static_cast(k); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1885 | return dk * ln2_hi - ((R - dk * ln2_lo) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1886 | } [ 120s] | ~ [ 120s] 1887 | } [ 120s] | ~ [ 120s] 1888 | s = f / (2.0 + f); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 1889 | dk = static_cast(k); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1890 | z = s * s; [ 120s] | ~~~~~~~~~~ [ 120s] 1891 | i = hx - 0x6147A; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1892 | w = z * z; [ 120s] | ~~~~~~~~~~ [ 120s] 1893 | j = 0x6B851 - hx; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 1894 | t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1895 | t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1896 | i |= j; [ 120s] | ~~~~~~~ [ 120s] 1897 | R = t2 + t1; [ 120s] | ~~~~~~~~~~~~ [ 120s] 1898 | if (i > 0) { [ 120s] | ~~~~~~~~~~~~ [ 120s] 1899 | hfsq = 0.5 * f * f; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 1900 | if (k == 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1901 | return f - (hfsq - s * (hfsq + R)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1902 | else [ 120s] | ~~~~ [ 120s] 1903 | return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1904 | } else { [ 120s] | ~~~~~~~~ [ 120s] 1905 | if (k == 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 1906 | return f - s * (f - R); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1907 | else [ 120s] | ~~~~ [ 120s] 1908 | return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1909 | } [ 120s] | ~ [ 120s] 1910 | } [ 120s] | ~ [ 120s] 1911 | [ 120s] | [ 120s] 1912 | /* double log1p(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1913 | * [ 120s] | ~ [ 120s] 1914 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 1915 | * 1. Argument Reduction: find k and f such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1916 | * 1+x = 2^k * (1+f), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1917 | * where sqrt(2)/2 < 1+f < sqrt(2) . [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1918 | * [ 120s] | ~ [ 120s] 1919 | * Note. If k=0, then f=x is exact. However, if k!=0, then f [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1920 | * may not be representable exactly. In that case, a correction [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1921 | * term is need. Let u=1+x rounded. Let c = (1+x)-u, then [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1922 | * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1923 | * and add back the correction term c/u. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1924 | * (Note: when x > 2**53, one can simply return log(x)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1925 | * [ 120s] | ~ [ 120s] 1926 | * 2. Approximation of log1p(f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1927 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1928 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1929 | * = 2s + s*R [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 1930 | * We use a special Reme algorithm on [0,0.1716] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1931 | * a polynomial of degree 14 to approximate R The maximum error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1932 | * of this polynomial approximation is bounded by 2**-58.45. In [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1933 | * other words, [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 1934 | * 2 4 6 8 10 12 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1935 | * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1936 | * (the values of Lp1 to Lp7 are listed in the program) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1937 | * and [ 120s] | ~~~~~~ [ 120s] 1938 | * | 2 14 | -58.45 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1939 | * | Lp1*s +...+Lp7*s - R(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1940 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1941 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1942 | * In order to guarantee error in log below 1ulp, we compute log [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1943 | * by [ 120s] | ~~~~~ [ 120s] 1944 | * log1p(f) = f - (hfsq - s*(hfsq+R)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1945 | * [ 120s] | ~ [ 120s] 1946 | * 3. Finally, log1p(x) = k*ln2 + log1p(f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1947 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1948 | * Here ln2 is split into two floating point number: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1949 | * ln2_hi + ln2_lo, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1950 | * where n*ln2_hi is always exact for |n| < 2000. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1951 | * [ 120s] | ~ [ 120s] 1952 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 1953 | * log1p(x) is NaN with signal if x < -1 (including -INF) ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1954 | * log1p(+INF) is +INF; log1p(-1) is -INF with signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1955 | * log1p(NaN) is that NaN with no signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1956 | * [ 120s] | ~ [ 120s] 1957 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 1958 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1959 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1960 | * [ 120s] | ~ [ 120s] 1961 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 1962 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1963 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1964 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1965 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1966 | * [ 120s] | ~ [ 120s] 1967 | * Note: Assuming log() return accurate answer, the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1968 | * algorithm can be used to compute log1p(x) to within a few ULP: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1969 | * [ 120s] | ~ [ 120s] 1970 | * u = 1+x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 1971 | * if(u==1.0) return x ; else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1972 | * return log(u)*(x/(u-1.0)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1973 | * [ 120s] | ~ [ 120s] 1974 | * See HP-15C Advanced Functions Handbook, p.193. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1975 | */ [ 120s] | ~~ [ 120s] 1976 | double log1p(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1977 | { [ 120s] | ~ [ 120s] 1978 | constexpr double /* -- */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1979 | ln2_hi [ 120s] | ~~~~~~ [ 120s] 1980 | = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1981 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1982 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1983 | Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1984 | Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1985 | Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1986 | Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1987 | Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1988 | Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1989 | Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1990 | [ 120s] | [ 120s] 1991 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1992 | constexpr volatile double vzero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1993 | [ 120s] | [ 120s] 1994 | double hfsq, f, c, s, z, R, u; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1995 | int32_t k, hx, hu, ax; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1996 | [ 120s] | [ 120s] 1997 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1998 | ax = hx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 1999 | [ 120s] | [ 120s] 2000 | c = 0; [ 120s] | ~~~~~~ [ 120s] 2001 | k = 1; [ 120s] | ~~~~~~ [ 120s] 2002 | if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2003 | if (ax >= 0x3FF00000) { /* x <= -1.0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2004 | if (x == -1.0) [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2005 | return -two54 / vzero; /* log1p(-1)=+inf */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2006 | else [ 120s] | ~~~~ [ 120s] 2007 | return (x - x) / (x - x); /* log1p(x<-1)=NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2008 | } [ 120s] | ~ [ 120s] 2009 | if (ax < 0x3E200000) { /* |x| < 2**-29 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2010 | if (two54 + x > zero /* raise inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2011 | && ax < 0x3C900000) /* |x| < 2**-54 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2012 | return x; [ 120s] | ~~~~~~~~~ [ 120s] 2013 | else [ 120s] | ~~~~ [ 120s] 2014 | return x - x * x * 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2015 | } [ 120s] | ~ [ 120s] 2016 | if (hx > 0 || hx <= static_cast(0xBFD2BEC4)) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2017 | k = 0; [ 120s] | ~~~~~~ [ 120s] 2018 | f = x; [ 120s] | ~~~~~~ [ 120s] 2019 | hu = 1; [ 120s] | ~~~~~~~ [ 120s] 2020 | } /* sqrt(2)/2- <= 1+x < sqrt(2)+ */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2021 | } [ 120s] | ~ [ 120s] 2022 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2023 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2024 | if (k != 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2025 | if (hx < 0x43400000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2026 | STRICT_ASSIGN(double, u, 1.0 + x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2027 | GET_HIGH_WORD(hu, u); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2028 | k = (hu >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2029 | c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2030 | c /= u; [ 120s] | ~~~~~~~ [ 120s] 2031 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2032 | u = x; [ 120s] | ~~~~~~ [ 120s] 2033 | GET_HIGH_WORD(hu, u); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2034 | k = (hu >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2035 | c = 0; [ 120s] | ~~~~~~ [ 120s] 2036 | } [ 120s] | ~ [ 120s] 2037 | hu &= 0x000FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2038 | /* [ 120s] | ~~ [ 120s] 2039 | * The approximation to sqrt(2) used in thresholds is not [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2040 | * critical. However, the ones used above must give less [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2041 | * strict bounds than the one here so that the k==0 case is [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2042 | * never reached from here, since here we have committed to [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2043 | * using the correction term but don't use it if k==0. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2044 | */ [ 120s] | ~~ [ 120s] 2045 | if (hu < 0x6A09E) { /* u ~< sqrt(2) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2046 | SET_HIGH_WORD(u, hu | 0x3FF00000); /* normalize u */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2047 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2048 | k += 1; [ 120s] | ~~~~~~~ [ 120s] 2049 | SET_HIGH_WORD(u, hu | 0x3FE00000); /* normalize u/2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2050 | hu = (0x00100000 - hu) >> 2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2051 | } [ 120s] | ~ [ 120s] 2052 | f = u - 1.0; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2053 | } [ 120s] | ~ [ 120s] 2054 | hfsq = 0.5 * f * f; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2055 | if (hu == 0) { /* |f| < 2**-20 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2056 | if (f == zero) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2057 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2058 | return zero; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2059 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2060 | c += k * ln2_lo; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2061 | return k * ln2_hi + c; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2062 | } [ 120s] | ~ [ 120s] 2063 | } [ 120s] | ~ [ 120s] 2064 | R = hfsq * (1.0 - 0.66666666666666666 * f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2065 | if (k == 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 2066 | return f - R; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2067 | else [ 120s] | ~~~~ [ 120s] 2068 | return k * ln2_hi - ((R - (k * ln2_lo + c)) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2069 | } [ 120s] | ~ [ 120s] 2070 | s = f / (2.0 + f); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2071 | z = s * s; [ 120s] | ~~~~~~~~~~ [ 120s] 2072 | R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7)))))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2073 | if (k == 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 2074 | return f - (hfsq - s * (hfsq + R)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2075 | else [ 120s] | ~~~~ [ 120s] 2076 | return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2077 | } [ 120s] | ~ [ 120s] 2078 | [ 120s] | [ 120s] 2079 | /* [ 120s] | ~~ [ 120s] 2080 | * k_log1p(f): [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2081 | * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)]. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2082 | * [ 120s] | ~ [ 120s] 2083 | * The following describes the overall strategy for computing [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2084 | * logarithms in base e. The argument reduction and adding the final [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2085 | * term of the polynomial are done by the caller for increased accuracy [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2086 | * when different bases are used. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2087 | * [ 120s] | ~ [ 120s] 2088 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2089 | * 1. Argument Reduction: find k and f such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2090 | * x = 2^k * (1+f), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2091 | * where sqrt(2)/2 < 1+f < sqrt(2) . [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2092 | * [ 120s] | ~ [ 120s] 2093 | * 2. Approximation of log(1+f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2094 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2095 | * = 2s + 2/3 s**3 + 2/5 s**5 + ....., [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2096 | * = 2s + s*R [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2097 | * We use a special Reme algorithm on [0,0.1716] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2098 | * a polynomial of degree 14 to approximate R The maximum error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2099 | * of this polynomial approximation is bounded by 2**-58.45. In [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2100 | * other words, [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2101 | * 2 4 6 8 10 12 14 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2102 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2103 | * (the values of Lg1 to Lg7 are listed in the program) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2104 | * and [ 120s] | ~~~~~~~~~~ [ 120s] 2105 | * | 2 14 | -58.45 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2106 | * | Lg1*s +...+Lg7*s - R(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2107 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2108 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2109 | * In order to guarantee error in log below 1ulp, we compute log [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2110 | * by [ 120s] | ~~~~~~~~~ [ 120s] 2111 | * log(1+f) = f - s*(f - R) (if f is not too large) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2112 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2113 | * [ 120s] | ~ [ 120s] 2114 | * 3. Finally, log(x) = k*ln2 + log(1+f). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2115 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2116 | * Here ln2 is split into two floating point number: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2117 | * ln2_hi + ln2_lo, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2118 | * where n*ln2_hi is always exact for |n| < 2000. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2119 | * [ 120s] | ~ [ 120s] 2120 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2121 | * log(x) is NaN with signal if x < 0 (including -INF) ; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2122 | * log(+INF) is +INF; log(0) is -INF with signal; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2123 | * log(NaN) is that NaN with no signal. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2124 | * [ 120s] | ~ [ 120s] 2125 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 2126 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2127 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2128 | * [ 120s] | ~ [ 120s] 2129 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 2130 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2131 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2132 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2133 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2134 | */ [ 120s] | ~~ [ 120s] 2135 | [ 120s] | [ 120s] 2136 | constexpr double Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2137 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2138 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2139 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2140 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2141 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2142 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2143 | [ 120s] | [ 120s] 2144 | /* [ 120s] | ~~ [ 120s] 2145 | * We always inline k_log1p(), since doing so produces a [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2146 | * substantial performance improvement (~40% on amd64). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2147 | */ [ 120s] | ~~ [ 120s] 2148 | static inline double k_log1p(double f) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2149 | { [ 120s] | ~ [ 120s] 2150 | double hfsq, s, z, R, w, t1, t2; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2151 | [ 120s] | [ 120s] 2152 | s = f / (2.0 + f); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2153 | z = s * s; [ 120s] | ~~~~~~~~~~ [ 120s] 2154 | w = z * z; [ 120s] | ~~~~~~~~~~ [ 120s] 2155 | t1 = w * (Lg2 + w * (Lg4 + w * Lg6)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2156 | t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2157 | R = t2 + t1; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2158 | hfsq = 0.5 * f * f; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2159 | return s * (hfsq + R); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2160 | } [ 120s] | ~ [ 120s] 2161 | [ 120s] | [ 120s] 2162 | /* [ 120s] | ~~ [ 120s] 2163 | * Return the base 2 logarithm of x. See e_log.c and k_log.h for most [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2164 | * comments. [ 120s] | ~~~~~~~~~~~ [ 120s] 2165 | * [ 120s] | ~ [ 120s] 2166 | * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2167 | * then does the combining and scaling steps [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2168 | * log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2169 | * in not-quite-routine extra precision. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2170 | */ [ 120s] | ~~ [ 120s] 2171 | double log2(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2172 | { [ 120s] | ~ [ 120s] 2173 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2174 | two54 [ 120s] | ~~~~~ [ 120s] 2175 | = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2176 | ivln2hi = 1.44269504072144627571e+00, /* 0x3FF71547, 0x65200000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2177 | ivln2lo = 1.67517131648865118353e-10; /* 0x3DE705FC, 0x2EEFA200 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2178 | [ 120s] | [ 120s] 2179 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2180 | constexpr volatile double vzero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2181 | [ 120s] | [ 120s] 2182 | double f, hfsq, hi, lo, r, val_hi, val_lo, w, y; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2183 | int32_t i, k, hx; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2184 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2185 | [ 120s] | [ 120s] 2186 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2187 | [ 120s] | [ 120s] 2188 | k = 0; [ 120s] | ~~~~~~ [ 120s] 2189 | if (hx < 0x00100000) { /* x < 2**-1022 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2190 | if (((hx & 0x7FFFFFFF) | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2191 | return -two54 / vzero; /* log(+-0)=-inf */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2192 | if (hx < 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 2193 | return (x - x) / zero; /* log(-#) = NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2194 | k -= 54; [ 120s] | ~~~~~~~~ [ 120s] 2195 | x *= two54; /* subnormal number, scale up x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2196 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2197 | } [ 120s] | ~ [ 120s] 2198 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2199 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2200 | if (hx == 0x3FF00000 && lx == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2201 | return zero; /* log(1) = +0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2202 | k += (hx >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2203 | hx &= 0x000FFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2204 | i = (hx + 0x95F64) & 0x100000; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2205 | SET_HIGH_WORD(x, hx | (i ^ 0x3FF00000)); /* normalize x or x/2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2206 | k += (i >> 20); [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2207 | y = static_cast(k); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2208 | f = x - 1.0; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2209 | hfsq = 0.5 * f * f; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2210 | r = k_log1p(f); [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2211 | [ 120s] | [ 120s] 2212 | /* [ 120s] | ~~ [ 120s] 2213 | * f-hfsq must (for args near 1) be evaluated in extra precision [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2214 | * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2215 | * This is fairly efficient since f-hfsq only depends on f, so can [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2216 | * be evaluated in parallel with R. Not combining hfsq with R also [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2217 | * keeps R small (though not as small as a true `lo' term would be), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2218 | * so that extra precision is not needed for terms involving R. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2219 | * [ 120s] | ~ [ 120s] 2220 | * Compiler bugs involving extra precision used to break Dekker's [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2221 | * theorem for spitting f-hfsq as hi+lo, unless double_t was used [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2222 | * or the multi-precision calculations were avoided when double_t [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2223 | * has extra precision. These problems are now automatically [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2224 | * avoided as a side effect of the optimization of combining the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2225 | * Dekker splitting step with the clear-low-bits step. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2226 | * [ 120s] | ~ [ 120s] 2227 | * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2228 | * precision to avoid a very large cancellation when x is very near [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2229 | * these values. Unlike the above cancellations, this problem is [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2230 | * specific to base 2. It is strange that adding +-1 is so much [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2231 | * harder than adding +-ln2 or +-log10_2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2232 | * [ 120s] | ~ [ 120s] 2233 | * This uses Dekker's theorem to normalize y+val_hi, so the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2234 | * compiler bugs are back in some configurations, sigh. And I [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2235 | * don't want to used double_t to avoid them, since that gives a [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2236 | * pessimization and the support for avoiding the pessimization [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2237 | * is not yet available. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2238 | * [ 120s] | ~ [ 120s] 2239 | * The multi-precision calculations for the multiplications are [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2240 | * routine. [ 120s] | ~~~~~~~~~~ [ 120s] 2241 | */ [ 120s] | ~~ [ 120s] 2242 | hi = f - hfsq; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2243 | SET_LOW_WORD(hi, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2244 | lo = (f - hi) - hfsq + r; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2245 | val_hi = hi * ivln2hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2246 | val_lo = (lo + hi) * ivln2lo + lo * ivln2hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2247 | [ 120s] | [ 120s] 2248 | /* spadd(val_hi, val_lo, y), except for not using double_t: */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2249 | w = y + val_hi; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2250 | val_lo += (y - w) + val_hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2251 | val_hi = w; [ 120s] | ~~~~~~~~~~~ [ 120s] 2252 | [ 120s] | [ 120s] 2253 | return val_lo + val_hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2254 | } [ 120s] | ~ [ 120s] 2255 | [ 120s] | [ 120s] 2256 | /* [ 120s] | ~~ [ 120s] 2257 | * Return the base 10 logarithm of x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2258 | * [ 120s] | ~ [ 120s] 2259 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2260 | * Let log10_2hi = leading 40 bits of log10(2) and [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2261 | * log10_2lo = log10(2) - log10_2hi, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2262 | * ivln10 = 1/log(10) rounded. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2263 | * Then [ 120s] | ~~~~~~~~~~~ [ 120s] 2264 | * n = ilogb(x), [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2265 | * if(n<0) n = n+1; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2266 | * x = scalbn(x,-n); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2267 | * log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2268 | * [ 120s] | ~ [ 120s] 2269 | * Note 1: [ 120s] | ~~~~~~~~~~ [ 120s] 2270 | * To guarantee log10(10**n)=n, where 10**n is normal, the rounding [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2271 | * mode must set to Round-to-Nearest. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2272 | * Note 2: [ 120s] | ~~~~~~~~~~ [ 120s] 2273 | * [1/log(10)] rounded to 53 bits has error .198 ulps; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2274 | * log10 is monotonic at all binary break points. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2275 | * [ 120s] | ~ [ 120s] 2276 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2277 | * log10(x) is NaN if x < 0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2278 | * log10(+INF) is +INF; log10(0) is -INF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2279 | * log10(NaN) is that NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2280 | * log10(10**N) = N for N=0,1,...,22. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2281 | */ [ 120s] | ~~ [ 120s] 2282 | double log10(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2283 | { [ 120s] | ~ [ 120s] 2284 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2285 | two54 [ 120s] | ~~~~~ [ 120s] 2286 | = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2287 | ivln10 = 4.34294481903251816668e-01, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2288 | log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2289 | log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2290 | [ 120s] | [ 120s] 2291 | constexpr double zero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2292 | constexpr volatile double vzero = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2293 | [ 120s] | [ 120s] 2294 | double y; [ 120s] | ~~~~~~~~~ [ 120s] 2295 | int32_t i, k, hx; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2296 | uint32_t lx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2297 | [ 120s] | [ 120s] 2298 | EXTRACT_WORDS(hx, lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2299 | [ 120s] | [ 120s] 2300 | k = 0; [ 120s] | ~~~~~~ [ 120s] 2301 | if (hx < 0x00100000) { /* x < 2**-1022 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2302 | if (((hx & 0x7FFFFFFF) | lx) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2303 | return -two54 / vzero; /* log(+-0)=-inf */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2304 | if (hx < 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 2305 | return (x - x) / zero; /* log(-#) = NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2306 | k -= 54; [ 120s] | ~~~~~~~~ [ 120s] 2307 | x *= two54; /* subnormal number, scale up x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2308 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2309 | GET_LOW_WORD(lx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2310 | } [ 120s] | ~ [ 120s] 2311 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2312 | return x + x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2313 | if (hx == 0x3FF00000 && lx == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2314 | return zero; /* log(1) = +0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2315 | k += (hx >> 20) - 1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2316 | [ 120s] | [ 120s] 2317 | i = (k & 0x80000000) >> 31; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2318 | hx = (hx & 0x000FFFFF) | ((0x3FF - i) << 20); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2319 | y = k + i; [ 120s] | ~~~~~~~~~~ [ 120s] 2320 | SET_HIGH_WORD(x, hx); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2321 | SET_LOW_WORD(x, lx); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2322 | [ 120s] | [ 120s] 2323 | double z = y * log10_2lo + ivln10 * log(x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2324 | return z + y * log10_2hi; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2325 | } [ 120s] | ~ [ 120s] 2326 | [ 120s] | [ 120s] 2327 | /* expm1(x) [ 120s] | ~~~~~~~~~~~ [ 120s] 2328 | * Returns exp(x)-1, the exponential of x minus 1. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2329 | * [ 120s] | ~ [ 120s] 2330 | * Method [ 120s] | ~~~~~~~~ [ 120s] 2331 | * 1. Argument reduction: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2332 | * Given x, find r and integer k such that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2333 | * [ 120s] | ~ [ 120s] 2334 | * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2335 | * [ 120s] | ~ [ 120s] 2336 | * Here a correction term c will be computed to compensate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2337 | * the error in r when rounded to a floating-point number. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2338 | * [ 120s] | ~ [ 120s] 2339 | * 2. Approximating expm1(r) by a special rational function on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2340 | * the interval [0,0.34658]: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2341 | * Since [ 120s] | ~~~~~~~~ [ 120s] 2342 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2343 | * we define R1(r*r) by [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2344 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2345 | * That is, [ 120s] | ~~~~~~~~~~~ [ 120s] 2346 | * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2347 | * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2348 | * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2349 | * We use a special Reme algorithm on [0,0.347] to generate [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2350 | * a polynomial of degree 5 in r*r to approximate R1. The [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2351 | * maximum error of this polynomial approximation is bounded [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2352 | * by 2**-61. In other words, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2353 | * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2354 | * where Q1 = -1.6666666666666567384E-2, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2355 | * Q2 = 3.9682539681370365873E-4, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2356 | * Q3 = -9.9206344733435987357E-6, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2357 | * Q4 = 2.5051361420808517002E-7, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2358 | * Q5 = -6.2843505682382617102E-9; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2359 | * z = r*r, [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2360 | * with error bounded by [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2361 | * | 5 | -61 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2362 | * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2363 | * | | [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2364 | * [ 120s] | ~ [ 120s] 2365 | * expm1(r) = exp(r)-1 is then computed by the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2366 | * specific way which minimize the accumulation rounding error: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2367 | * 2 3 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2368 | * r r [ 3 - (R1 + R1*r/2) ] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2369 | * expm1(r) = r + --- + --- * [--------------------] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2370 | * 2 2 [ 6 - r*(3 - R1*r/2) ] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2371 | * [ 120s] | ~ [ 120s] 2372 | * To compensate the error in the argument reduction, we use [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2373 | * expm1(r+c) = expm1(r) + c + expm1(r)*c [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2374 | * ~ expm1(r) + c + r*c [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2375 | * Thus c+r*c will be added in as the correction terms for [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2376 | * expm1(r+c). Now rearrange the term to avoid optimization [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2377 | * screw up: [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2378 | * ( 2 2 ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2379 | * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2380 | * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2381 | * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2382 | * ( ) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2383 | * [ 120s] | ~ [ 120s] 2384 | * = r - E [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2385 | * 3. Scale back to obtain expm1(x): [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2386 | * From step 1, we have [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2387 | * expm1(x) = either 2^k*[expm1(r)+1] - 1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2388 | * = or 2^k*[expm1(r) + (1-2^-k)] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2389 | * 4. Implementation notes: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2390 | * (A). To save one multiplication, we scale the coefficient Qi [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2391 | * to Qi*2^i, and replace z by (x^2)/2. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2392 | * (B). To achieve maximum accuracy, we compute expm1(x) by [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2393 | * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2394 | * (ii) if k=0, return r-E [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2395 | * (iii) if k=-1, return 0.5*(r-E)-0.5 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2396 | * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2397 | * else return 1.0+2.0*(r-E); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2398 | * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2399 | * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2400 | * (vii) return 2^k(1-((E+2^-k)-r)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2401 | * [ 120s] | ~ [ 120s] 2402 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2403 | * expm1(INF) is INF, expm1(NaN) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2404 | * expm1(-INF) is -1, and [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2405 | * for finite argument, only expm1(0)=0 is exact. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2406 | * [ 120s] | ~ [ 120s] 2407 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 2408 | * according to an error analysis, the error is always less than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2409 | * 1 ulp (unit in the last place). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2410 | * [ 120s] | ~ [ 120s] 2411 | * Misc. info. [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2412 | * For IEEE double [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2413 | * if x > 7.09782712893383973096e+02 then expm1(x) overflow [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2414 | * [ 120s] | ~ [ 120s] 2415 | * Constants: [ 120s] | ~~~~~~~~~~~~ [ 120s] 2416 | * The hexadecimal values are the intended ones for the following [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2417 | * constants. The decimal values may be used, provided that the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2418 | * compiler will convert from decimal to binary accurately enough [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2419 | * to produce the hexadecimal values shown. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2420 | */ [ 120s] | ~~ [ 120s] 2421 | double expm1(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2422 | { [ 120s] | ~ [ 120s] 2423 | constexpr double [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2424 | one [ 120s] | ~~~ [ 120s] 2425 | = 1.0, [ 120s] | ~~~~~~ [ 120s] 2426 | tiny = 1.0e-300, [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2427 | o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2428 | ln2_hi = 6.93147180369123816490e-01, /* 0x3FE62E42, 0xFEE00000 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2429 | ln2_lo = 1.90821492927058770002e-10, /* 0x3DEA39EF, 0x35793C76 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2430 | invln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2431 | /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2432 | x*x/2: */ [ 120s] | ~~~~~~~~~ [ 120s] 2433 | Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2434 | Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2435 | Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2436 | Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2437 | Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2438 | [ 120s] | [ 120s] 2439 | constexpr volatile double huge = 1.0e+300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2440 | [ 120s] | [ 120s] 2441 | double y, hi, lo, c, t, e, hxs, hfx, r1, twopk; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2442 | int32_t k, xsb; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2443 | uint32_t hx; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2444 | [ 120s] | [ 120s] 2445 | GET_HIGH_WORD(hx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2446 | xsb = hx & 0x80000000; /* sign bit of x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2447 | hx &= 0x7FFFFFFF; /* high word of |x| */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2448 | [ 120s] | [ 120s] 2449 | /* filter out huge and non-finite argument */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2450 | if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2451 | if (hx >= 0x40862E42) { /* if |x|>=709.78... */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2452 | if (hx >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2453 | uint32_t low; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2454 | GET_LOW_WORD(low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2455 | if (((hx & 0xFFFFF) | low) != 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2456 | return x + x; /* NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2457 | else [ 120s] | ~~~~ [ 120s] 2458 | return (xsb == 0) ? x : -1.0; /* exp(+-inf)={inf,-1} */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2459 | } [ 120s] | ~ [ 120s] 2460 | if (x > o_threshold) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2461 | return huge * huge; /* overflow */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2462 | } [ 120s] | ~ [ 120s] 2463 | if (xsb != 0 && x + tiny < 0.0) { /* x < -56*ln2, return -1.0 with inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2464 | /* raise inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2465 | return tiny - one; /* return -1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2466 | } [ 120s] | ~ [ 120s] 2467 | } [ 120s] | ~ [ 120s] 2468 | [ 120s] | [ 120s] 2469 | /* argument reduction */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2470 | if (hx > 0x3FD62E42) { /* if |x| > 0.5 ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2471 | if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2472 | if (xsb == 0) { [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2473 | hi = x - ln2_hi; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2474 | lo = ln2_lo; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2475 | k = 1; [ 120s] | ~~~~~~ [ 120s] 2476 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2477 | hi = x + ln2_hi; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2478 | lo = -ln2_lo; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2479 | k = -1; [ 120s] | ~~~~~~~ [ 120s] 2480 | } [ 120s] | ~ [ 120s] 2481 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2482 | k = invln2 * x + ((xsb == 0) ? 0.5 : -0.5); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2483 | t = k; [ 120s] | ~~~~~~ [ 120s] 2484 | hi = x - t * ln2_hi; /* t*ln2_hi is exact here */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2485 | lo = t * ln2_lo; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2486 | } [ 120s] | ~ [ 120s] 2487 | STRICT_ASSIGN(double, x, hi - lo); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2488 | c = (hi - x) - lo; [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2489 | } else if (hx < 0x3C900000) { /* when |x|<2**-54, return x */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2490 | t = huge + x; /* return x with inexact flags when x!=0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2491 | return x - (t - (huge + x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2492 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2493 | k = 0; [ 120s] | ~~~~~~ [ 120s] 2494 | } [ 120s] | ~ [ 120s] 2495 | [ 120s] | [ 120s] 2496 | /* x is now in primary range */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2497 | hfx = 0.5 * x; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2498 | hxs = x * hfx; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2499 | r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5)))); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2500 | t = 3.0 - r1 * hfx; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2501 | e = hxs * ((r1 - t) / (6.0 - x * t)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2502 | if (k == 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2503 | return x - (x * e - hxs); /* c is 0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2504 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2505 | INSERT_WORDS(twopk, 0x3FF00000 + (k << 20), 0); /* 2^k */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2506 | e = (x * (e - c) - c); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2507 | e -= hxs; [ 120s] | ~~~~~~~~~ [ 120s] 2508 | if (k == -1) [ 120s] | ~~~~~~~~~~~~ [ 120s] 2509 | return 0.5 * (x - e) - 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2510 | if (k == 1) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2511 | if (x < -0.25) [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2512 | return -2.0 * (e - (x + 0.5)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2513 | else [ 120s] | ~~~~ [ 120s] 2514 | return one + 2.0 * (x - e); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2515 | } [ 120s] | ~ [ 120s] 2516 | if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2517 | y = one - (e - x); [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2518 | // TODO(mvstanton): is this replacement for the hex float [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2519 | // sufficient? [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2520 | // if (k == 1024) y = y*2.0*0x1p1023; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2521 | if (k == 1024) [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2522 | y = y * 2.0 * 8.98846567431158e+307; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2523 | else [ 120s] | ~~~~ [ 120s] 2524 | y = y * twopk; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2525 | return y - one; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2526 | } [ 120s] | ~ [ 120s] 2527 | t = one; [ 120s] | ~~~~~~~~ [ 120s] 2528 | if (k < 20) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2529 | SET_HIGH_WORD(t, 0x3FF00000 - (0x200000 >> k)); /* t=1-2^-k */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2530 | y = t - (e - x); [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2531 | y = y * twopk; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2532 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2533 | SET_HIGH_WORD(t, ((0x3FF - k) << 20)); /* 2^-k */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2534 | y = x - (e + t); [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2535 | y += one; [ 120s] | ~~~~~~~~~ [ 120s] 2536 | y = y * twopk; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2537 | } [ 120s] | ~ [ 120s] 2538 | } [ 120s] | ~ [ 120s] 2539 | return y; [ 120s] | ~~~~~~~~~ [ 120s] 2540 | } [ 120s] | ~ [ 120s] 2541 | [ 120s] | [ 120s] 2542 | double cbrt(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2543 | { [ 120s] | ~ [ 120s] 2544 | constexpr uint32_t [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2545 | B1 [ 120s] | ~~ [ 120s] 2546 | = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2547 | B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2548 | [ 120s] | [ 120s] 2549 | /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2550 | constexpr double P0 = 1.87595182427177009643, /* 0x3FFE03E6, 0x0F61E692 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2551 | P1 = -1.88497979543377169875, /* 0xBFFE28E0, 0x92F02420 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2552 | P2 = 1.621429720105354466140, /* 0x3FF9F160, 0x4A49D6C2 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2553 | P3 = -0.758397934778766047437, /* 0xBFE844CB, 0xBEE751D9 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2554 | P4 = 0.145996192886612446982; /* 0x3FC2B000, 0xD4E4EDD7 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2555 | [ 120s] | [ 120s] 2556 | int32_t hx; [ 120s] | ~~~~~~~~~~~ [ 120s] 2557 | union { [ 120s] | ~~~~~~~ [ 120s] 2558 | double value; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2559 | uint64_t bits; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2560 | } u; [ 120s] | ~~~~ [ 120s] 2561 | double r, s, t = 0.0, w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2562 | uint32_t sign; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2563 | uint32_t high, low; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2564 | [ 120s] | [ 120s] 2565 | EXTRACT_WORDS(hx, low, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2566 | sign = hx & 0x80000000; /* sign= sign(x) */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2567 | hx ^= sign; [ 120s] | ~~~~~~~~~~~ [ 120s] 2568 | if (hx >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2569 | return (x + x); /* cbrt(NaN,INF) is itself */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2570 | [ 120s] | [ 120s] 2571 | /* [ 120s] | ~~ [ 120s] 2572 | * Rough cbrt to 5 bits: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2573 | * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2574 | * where e is integral and >= 0, m is real and in [0, 1), and "/" and [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2575 | * "%" are integer division and modulus with rounding towards minus [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2576 | * infinity. The RHS is always >= the LHS and has a maximum relative [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2577 | * error of about 1 in 16. Adding a bias of -0.03306235651 to the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2578 | * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2579 | * floating point representation, for finite positive normal values, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2580 | * ordinary integer division of the value in bits magically gives [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2581 | * almost exactly the RHS of the above provided we first subtract the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2582 | * exponent bias (1023 for doubles) and later add it back. We do the [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2583 | * subtraction virtually to keep e >= 0 so that ordinary integer [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2584 | * division rounds towards minus infinity; this is also efficient. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2585 | */ [ 120s] | ~~ [ 120s] 2586 | if (hx < 0x00100000) { /* zero or subnormal? */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2587 | if ((hx | low) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2588 | return (x); /* cbrt(0) is itself */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2589 | SET_HIGH_WORD(t, 0x43500000); /* set t= 2**54 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2590 | t *= x; [ 120s] | ~~~~~~~ [ 120s] 2591 | GET_HIGH_WORD(high, t); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2592 | INSERT_WORDS(t, sign | ((high & 0x7FFFFFFF) / 3 + B2), 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2593 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2594 | INSERT_WORDS(t, sign | (hx / 3 + B1), 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2595 | } [ 120s] | ~ [ 120s] 2596 | [ 120s] | [ 120s] 2597 | /* [ 120s] | ~~ [ 120s] 2598 | * New cbrt to 23 bits: [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2599 | * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2600 | * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2601 | * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2602 | * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2603 | * gives us bounds for r = t**3/x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2604 | * [ 120s] | ~ [ 120s] 2605 | * Try to optimize for parallel evaluation as in k_tanf.c. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2606 | */ [ 120s] | ~~ [ 120s] 2607 | r = (t * t) * (t / x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2608 | t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2609 | [ 120s] | [ 120s] 2610 | /* [ 120s] | ~~ [ 120s] 2611 | * Round t away from zero to 23 bits (sloppily except for ensuring that [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2612 | * the result is larger in magnitude than cbrt(x) but not much more than [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2613 | * 2 23-bit ulps larger). With rounding towards zero, the error bound [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2614 | * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2615 | * in the rounded t, the infinite-precision error in the Newton [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2616 | * approximation barely affects third digit in the final error [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2617 | * 0.667; the error in the rounded t can be up to about 3 23-bit ulps [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2618 | * before the final error is larger than 0.667 ulps. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2619 | */ [ 120s] | ~~ [ 120s] 2620 | u.value = t; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2621 | u.bits = (u.bits + 0x80000000) & 0xFFFFFFFFC0000000ULL; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2622 | t = u.value; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2623 | [ 120s] | [ 120s] 2624 | /* one step Newton iteration to 53 bits with error < 0.667 ulps */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2625 | s = t * t; /* t*t is exact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2626 | r = x / s; /* error <= 0.5 ulps; |r| < |t| */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2627 | w = t + t; /* t+t is exact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2628 | r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2629 | t = t + t * r; /* error <= 0.5 + 0.5/3 + epsilon */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2630 | [ 120s] | [ 120s] 2631 | return (t); [ 120s] | ~~~~~~~~~~~ [ 120s] 2632 | } [ 120s] | ~ [ 120s] 2633 | [ 120s] | [ 120s] 2634 | /* sin(x) [ 120s] | ~~~~~~~~~ [ 120s] 2635 | * Return sine function of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2636 | * [ 120s] | ~ [ 120s] 2637 | * kernel function: [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2638 | * __kernel_sin ... sine function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2639 | * __kernel_cos ... cose function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2640 | * __ieee754_rem_pio2 ... argument reduction routine [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2641 | * [ 120s] | ~ [ 120s] 2642 | * Method. [ 120s] | ~~~~~~~~~ [ 120s] 2643 | * Let S,C and T denote the sin, cos and tan respectively on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2644 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2645 | * in [-pi/4 , +pi/4], and let n = k mod 4. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2646 | * We have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2647 | * [ 120s] | ~ [ 120s] 2648 | * n sin(x) cos(x) tan(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2649 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2650 | * 0 S C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2651 | * 1 C -S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2652 | * 2 -S -C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2653 | * 3 -C S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2654 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2655 | * [ 120s] | ~ [ 120s] 2656 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2657 | * Let trig be any of sin, cos, or tan. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2658 | * trig(+-INF) is NaN, with signals; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2659 | * trig(NaN) is that NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2660 | * [ 120s] | ~ [ 120s] 2661 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 2662 | * TRIG(x) returns trig(x) nearly rounded [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2663 | */ [ 120s] | ~~ [ 120s] 2664 | double sin(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2665 | { [ 120s] | ~ [ 120s] 2666 | double y[2], z = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2667 | int32_t n, ix; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2668 | [ 120s] | [ 120s] 2669 | /* High word of x. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2670 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2671 | [ 120s] | [ 120s] 2672 | /* |x| ~< pi/4 */ [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2673 | ix &= 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2674 | if (ix <= 0x3FE921FB) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2675 | return __kernel_sin(x, z, 0); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2676 | } else if (ix >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2677 | /* sin(Inf or NaN) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2678 | return x - x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2679 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2680 | /* argument reduction needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2681 | n = __ieee754_rem_pio2(x, y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2682 | switch (n & 3) { [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2683 | case 0: [ 120s] | ~~~~~~~ [ 120s] 2684 | return __kernel_sin(y[0], y[1], 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2685 | case 1: [ 120s] | ~~~~~~~ [ 120s] 2686 | return __kernel_cos(y[0], y[1]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2687 | case 2: [ 120s] | ~~~~~~~ [ 120s] 2688 | return -__kernel_sin(y[0], y[1], 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2689 | default: [ 120s] | ~~~~~~~~ [ 120s] 2690 | return -__kernel_cos(y[0], y[1]); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2691 | } [ 120s] | ~ [ 120s] 2692 | } [ 120s] | ~ [ 120s] 2693 | } [ 120s] | ~ [ 120s] 2694 | [ 120s] | [ 120s] 2695 | /* tan(x) [ 120s] | ~~~~~~~~~ [ 120s] 2696 | * Return tangent function of x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2697 | * [ 120s] | ~ [ 120s] 2698 | * kernel function: [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2699 | * __kernel_tan ... tangent function on [-pi/4,pi/4] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2700 | * __ieee754_rem_pio2 ... argument reduction routine [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2701 | * [ 120s] | ~ [ 120s] 2702 | * Method. [ 120s] | ~~~~~~~~~ [ 120s] 2703 | * Let S,C and T denote the sin, cos and tan respectively on [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2704 | * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2705 | * in [-pi/4 , +pi/4], and let n = k mod 4. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2706 | * We have [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2707 | * [ 120s] | ~ [ 120s] 2708 | * n sin(x) cos(x) tan(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2709 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2710 | * 0 S C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2711 | * 1 C -S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2712 | * 2 -S -C T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2713 | * 3 -C S -1/T [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2714 | * ---------------------------------------------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2715 | * [ 120s] | ~ [ 120s] 2716 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2717 | * Let trig be any of sin, cos, or tan. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2718 | * trig(+-INF) is NaN, with signals; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2719 | * trig(NaN) is that NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2720 | * [ 120s] | ~ [ 120s] 2721 | * Accuracy: [ 120s] | ~~~~~~~~~~~ [ 120s] 2722 | * TRIG(x) returns trig(x) nearly rounded [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2723 | */ [ 120s] | ~~ [ 120s] 2724 | double tan(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2725 | { [ 120s] | ~ [ 120s] 2726 | double y[2], z = 0.0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2727 | int32_t n, ix; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2728 | [ 120s] | [ 120s] 2729 | /* High word of x. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2730 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2731 | [ 120s] | [ 120s] 2732 | /* |x| ~< pi/4 */ [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2733 | ix &= 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2734 | if (ix <= 0x3FE921FB) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2735 | return __kernel_tan(x, z, 1); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2736 | } else if (ix >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2737 | /* tan(Inf or NaN) is NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2738 | return x - x; /* NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2739 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2740 | /* argument reduction needed */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2741 | n = __ieee754_rem_pio2(x, y); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2742 | /* 1 -> n even, -1 -> n odd */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2743 | return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2744 | } [ 120s] | ~ [ 120s] 2745 | } [ 120s] | ~ [ 120s] 2746 | [ 120s] | [ 120s] 2747 | /* [ 120s] | ~~ [ 120s] 2748 | * ES6 draft 09-27-13, section 20.2.2.12. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2749 | * Math.cosh [ 120s] | ~~~~~~~~~~~ [ 120s] 2750 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2751 | * mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2752 | * 1. Replace x by |x| (cosh(x) = cosh(-x)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2753 | * 2. [ 120s] | ~~~~~~~~~ [ 120s] 2754 | * [ exp(x) - 1 ]^2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2755 | * 0 <= x <= ln2/2 : cosh(x) := 1 + ------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2756 | * 2*exp(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2757 | * [ 120s] | ~ [ 120s] 2758 | * exp(x) + 1/exp(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2759 | * ln2/2 <= x <= 22 : cosh(x) := ------------------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2760 | * 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2761 | * 22 <= x <= lnovft : cosh(x) := exp(x)/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2762 | * lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2763 | * ln2ovft < x : cosh(x) := huge*huge (overflow) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2764 | * [ 120s] | ~ [ 120s] 2765 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2766 | * cosh(x) is |x| if x is +INF, -INF, or NaN. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2767 | * only cosh(0)=1 is exact for finite x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2768 | */ [ 120s] | ~~ [ 120s] 2769 | double cosh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2770 | { [ 120s] | ~ [ 120s] 2771 | constexpr double KCOSH_OVERFLOW = 710.4758600739439; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2772 | constexpr double one = 1.0, half = 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2773 | constexpr volatile double huge = 1.0e+300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2774 | [ 120s] | [ 120s] 2775 | int32_t ix; [ 120s] | ~~~~~~~~~~~ [ 120s] 2776 | [ 120s] | [ 120s] 2777 | /* High word of |x|. */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2778 | GET_HIGH_WORD(ix, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2779 | ix &= 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2780 | [ 120s] | [ 120s] 2781 | // |x| in [0,0.5*log2], return 1+expm1(|x|)^2/(2*exp(|x|)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2782 | if (ix < 0x3FD62E43) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2783 | double t = expm1(fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2784 | double w = one + t; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2785 | // For |x| < 2^-55, cosh(x) = 1 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2786 | if (ix < 0x3C800000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2787 | return w; [ 120s] | ~~~~~~~~~ [ 120s] 2788 | return one + (t * t) / (w + w); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2789 | } [ 120s] | ~ [ 120s] 2790 | [ 120s] | [ 120s] 2791 | // |x| in [0.5*log2, 22], return (exp(|x|)+1/exp(|x|)/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2792 | if (ix < 0x40360000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2793 | double t = exp(fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2794 | return half * t + half / t; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2795 | } [ 120s] | ~ [ 120s] 2796 | [ 120s] | [ 120s] 2797 | // |x| in [22, log(maxdouble)], return half*exp(|x|) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2798 | if (ix < 0x40862E42) [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2799 | return half * exp(fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2800 | [ 120s] | [ 120s] 2801 | // |x| in [log(maxdouble), overflowthreshold] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2802 | if (fabs(x) <= KCOSH_OVERFLOW) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2803 | double w = exp(half * fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2804 | double t = half * w; [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2805 | return t * w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2806 | } [ 120s] | ~ [ 120s] 2807 | [ 120s] | [ 120s] 2808 | /* x is INF or NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2809 | if (ix >= 0x7FF00000) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2810 | return x * x; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2811 | [ 120s] | [ 120s] 2812 | // |x| > overflowthreshold. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2813 | return huge * huge; [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2814 | } [ 120s] | ~ [ 120s] 2815 | [ 120s] | [ 120s] 2816 | /* [ 120s] | ~~ [ 120s] 2817 | * ES6 draft 09-27-13, section 20.2.2.30. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2818 | * Math.sinh [ 120s] | ~~~~~~~~~~~ [ 120s] 2819 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2820 | * mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2821 | * 1. Replace x by |x| (sinh(-x) = -sinh(x)). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2822 | * 2. [ 120s] | ~~~~~~~~~ [ 120s] 2823 | * E + E/(E+1) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2824 | * 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2825 | * 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2826 | * [ 120s] | ~ [ 120s] 2827 | * 22 <= x <= lnovft : sinh(x) := exp(x)/2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2828 | * lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2829 | * ln2ovft < x : sinh(x) := x*shuge (overflow) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2830 | * [ 120s] | ~ [ 120s] 2831 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2832 | * sinh(x) is |x| if x is +Infinity, -Infinity, or NaN. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2833 | * only sinh(0)=0 is exact for finite x. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2834 | */ [ 120s] | ~~ [ 120s] 2835 | double sinh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2836 | { [ 120s] | ~ [ 120s] 2837 | constexpr double KSINH_OVERFLOW = 710.4758600739439, [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2838 | TWO_M28 = 3.725290298461914e-9, // 2^-28, empty lower half [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2839 | LOG_MAXD = 709.7822265625; // 0x40862E42 00000000, empty lower half [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2840 | constexpr double shuge = 1.0e307; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2841 | [ 120s] | [ 120s] 2842 | double h = (x < 0) ? -0.5 : 0.5; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2843 | // |x| in [0, 22]. return sign(x)*0.5*(E+E/(E+1)) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2844 | double ax = fabs(x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~ [ 120s] 2845 | if (ax < 22) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2846 | // For |x| < 2^-28, sinh(x) = x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2847 | if (ax < TWO_M28) [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2848 | return x; [ 120s] | ~~~~~~~~~ [ 120s] 2849 | double t = expm1(ax); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2850 | if (ax < 1) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2851 | return h * (2 * t - t * t / (t + 1)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2852 | } [ 120s] | ~ [ 120s] 2853 | return h * (t + t / (t + 1)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2854 | } [ 120s] | ~ [ 120s] 2855 | // |x| in [22, log(maxdouble)], return 0.5 * exp(|x|) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2856 | if (ax < LOG_MAXD) [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2857 | return h * exp(ax); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2858 | // |x| in [log(maxdouble), overflowthreshold] [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2859 | // overflowthreshold = 710.4758600739426 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2860 | if (ax <= KSINH_OVERFLOW) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2861 | double w = exp(0.5 * ax); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2862 | double t = h * w; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2863 | return t * w; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2864 | } [ 120s] | ~ [ 120s] 2865 | // |x| > overflowthreshold or is NaN. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2866 | // Return Infinity of the appropriate sign or NaN. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2867 | return x * shuge; [ 120s] | ~~~~~~~~~~~~~~~~~ [ 120s] 2868 | } [ 120s] | ~ [ 120s] 2869 | [ 120s] | [ 120s] 2870 | /* Tanh(x) [ 120s] | ~~~~~~~~~~ [ 120s] 2871 | * Return the Hyperbolic Tangent of x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2872 | * [ 120s] | ~ [ 120s] 2873 | * Method : [ 120s] | ~~~~~~~~~~ [ 120s] 2874 | * x -x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2875 | * e - e [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2876 | * 0. tanh(x) is defined to be ----------- [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2877 | * x -x [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2878 | * e + e [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2879 | * 1. reduce x to non-negative by tanh(-x) = -tanh(x). [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2880 | * 2. 0 <= x < 2**-28 : tanh(x) := x with inexact if x != 0 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2881 | * -t [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2882 | * 2**-28 <= x < 1 : tanh(x) := -----; t = expm1(-2x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2883 | * t + 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2884 | * 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2885 | * 1 <= x < 22 : tanh(x) := 1 - -----; t = expm1(2x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2886 | * t + 2 [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2887 | * 22 <= x <= INF : tanh(x) := 1. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2888 | * [ 120s] | ~ [ 120s] 2889 | * Special cases: [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2890 | * tanh(NaN) is NaN; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2891 | * only tanh(0)=0 is exact for finite argument. [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2892 | */ [ 120s] | ~~ [ 120s] 2893 | double tanh(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2894 | { [ 120s] | ~ [ 120s] 2895 | constexpr volatile double tiny = 1.0e-300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2896 | constexpr double one = 1.0, two = 2.0, huge = 1.0e300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2897 | double t, z; [ 120s] | ~~~~~~~~~~~~ [ 120s] 2898 | int32_t jx, ix; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2899 | [ 120s] | [ 120s] 2900 | GET_HIGH_WORD(jx, x); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2901 | ix = jx & 0x7FFFFFFF; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2902 | [ 120s] | [ 120s] 2903 | /* x is INF or NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2904 | if (ix >= 0x7FF00000) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2905 | if (jx >= 0) [ 120s] | ~~~~~~~~~~~~ [ 120s] 2906 | return one / x + one; /* tanh(+-inf)=+-1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2907 | else [ 120s] | ~~~~ [ 120s] 2908 | return one / x - one; /* tanh(NaN) = NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2909 | } [ 120s] | ~ [ 120s] 2910 | [ 120s] | [ 120s] 2911 | /* |x| < 22 */ [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2912 | if (ix < 0x40360000) { /* |x|<22 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2913 | if (ix < 0x3E300000 && huge + x > one) { /* |x|<2**-28 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2914 | return x; /* tanh(tiny) = tiny with inexact */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2915 | } [ 120s] | ~ [ 120s] 2916 | if (ix >= 0x3FF00000) { /* |x|>=1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2917 | t = expm1(two * fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2918 | z = one - two / (t + two); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2919 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2920 | t = expm1(-two * fabs(x)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2921 | z = -t / (t + two); [ 120s] | ~~~~~~~~~~~~~~~~~~~ [ 120s] 2922 | } [ 120s] | ~ [ 120s] 2923 | /* |x| >= 22, return +-1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2924 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2925 | z = one - tiny; /* raise inexact flag */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2926 | } [ 120s] | ~ [ 120s] 2927 | return (jx >= 0) ? z : -z; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2928 | } [ 120s] | ~ [ 120s] 2929 | [ 120s] | [ 120s] 2930 | double ceil(double x) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2931 | { [ 120s] | ~ [ 120s] 2932 | constexpr double huge = 1.0e300; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2933 | int i0, i1, j0; [ 120s] | ~~~~~~~~~~~~~~~ [ 120s] 2934 | unsigned i, j; [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2935 | i0 = __HI(x); [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2936 | i1 = __LO(x); [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2937 | j0 = ((i0 >> 20) & 0x7ff) - 0x3ff; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2938 | if (j0 < 20) { [ 120s] | ~~~~~~~~~~~~~~ [ 120s] 2939 | if (j0 < 0) { /* raise inexact if x != 0 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2940 | if (huge + x > 0.0) { /* return 0*sign(x) if |x|<1 */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2941 | if (i0 < 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2942 | i0 = 0x80000000; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2943 | i1 = 0; [ 120s] | ~~~~~~~ [ 120s] 2944 | } else if ((i0 | i1) != 0) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2945 | i0 = 0x3ff00000; [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2946 | i1 = 0; [ 120s] | ~~~~~~~ [ 120s] 2947 | } [ 120s] | ~ [ 120s] 2948 | } [ 120s] | ~ [ 120s] 2949 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2950 | i = (0x000fffff) >> j0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2951 | if (((i0 & i) | i1) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2952 | return x; /* x is integral */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2953 | if (huge + x > 0.0) { /* raise inexact flag */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2954 | if (i0 > 0) [ 120s] | ~~~~~~~~~~~ [ 120s] 2955 | i0 += (0x00100000) >> j0; [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2956 | i0 &= (~i); [ 120s] | ~~~~~~~~~~~ [ 120s] 2957 | i1 = 0; [ 120s] | ~~~~~~~ [ 120s] 2958 | } [ 120s] | ~ [ 120s] 2959 | } [ 120s] | ~ [ 120s] 2960 | } else if (j0 > 51) { [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2961 | if (j0 == 0x400) [ 120s] | ~~~~~~~~~~~~~~~~ [ 120s] 2962 | return x + x; /* inf or NaN */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2963 | else [ 120s] | ~~~~ [ 120s] 2964 | return x; /* x is integral */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2965 | } else { [ 120s] | ~~~~~~~~ [ 120s] 2966 | i = ((unsigned)(0xffffffff)) >> (j0 - 20); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2967 | if ((i1 & i) == 0) [ 120s] | ~~~~~~~~~~~~~~~~~~ [ 120s] 2968 | return x; /* x is integral */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2969 | if (huge + x > 0.0) { /* raise inexact flag */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2970 | if (i0 > 0) { [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2971 | if (j0 == 20) [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2972 | i0 += 1; [ 120s] | ~~~~~~~~ [ 120s] 2973 | else { [ 120s] | ~~~~~~ [ 120s] 2974 | j = i1 + (1 << (52 - j0)); [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2975 | if (j < (unsigned)i1) [ 120s] | ~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2976 | i0 += 1; /* got a carry */ [ 120s] | ~~~~~~~~~~~~~~~~~~~~~~~~~~ [ 120s] 2977 | i1 = j; [ 120s] | ~~~~~~~ [ 120s] 2978 | } [ 120s] | ~ [ 120s] 2979 | } [ 120s] | ~ [ 120s] 2980 | i1 &= (~i); [ 120s] | ~~~~~~~~~~~ [ 120s] 2981 | } [ 120s] | ~ [ 120s] 2982 | } [ 120s] | ~ [ 120s] 2983 | __HI(x) = i0; [ 120s] | ~~~~~~~~~~~~~ [ 120s] 2984 | __LO(x) = i1; [ 120s] | ~~~~~~ [ 120s] 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deps/v8/tools/csvparser.js (stored 0%) [ 128s] adding: deps/v8/tools/profile.js (stored 0%) [ 128s] adding: deps/v8/tools/profile_view.js (stored 0%) [ 128s] adding: deps/v8/tools/logreader.js (stored 0%) [ 128s] adding: deps/v8/tools/arguments.js (stored 0%) [ 128s] adding: deps/v8/tools/tickprocessor.js (stored 0%) [ 128s] adding: deps/v8/tools/SourceMap.js (stored 0%) [ 128s] adding: deps/v8/tools/tickprocessor-driver.js (stored 0%) [ 128s] adding: deps/node-inspect/lib/_inspect.js (stored 0%) [ 128s] adding: deps/node-inspect/lib/internal/inspect_client.js (stored 0%) [ 128s] adding: deps/node-inspect/lib/internal/inspect_repl.js (stored 0%) [ 128s] adding: deps/acorn/acorn/dist/acorn.js (stored 0%) [ 128s] adding: deps/acorn/acorn-walk/dist/walk.js (stored 0%) [ 128s] adding: deps/acorn-plugins/acorn-class-fields/index.js (stored 0%) [ 128s] adding: deps/acorn-plugins/acorn-numeric-separator/index.js (stored 0%) [ 128s] adding: deps/acorn-plugins/acorn-private-class-elements/index.js (stored 0%) [ 128s] adding: deps/acorn-plugins/acorn-private-methods/index.js (stored 0%) [ 128s] adding: deps/acorn-plugins/acorn-static-class-features/index.js (stored 0%) [ 128s] adding: deps/cjs-module-lexer/lexer.js (stored 0%) [ 128s] adding: deps/cjs-module-lexer/dist/lexer.js (stored 0%) [ 129s] [133/231] ACTION liblwnode: node_js2c_16c8176d30d05044f5cbb72c692d4563 [ 129s] [134/231] STAMP obj/liblwnode.actions_rules_copies.stamp [ 129s] [135/231] CXX obj/src/liblwnode.node_i18n.o [ 130s] [136/231] CXX obj/src/liblwnode.json_utils.o [ 131s] [137/231] CXX obj/src/liblwnode.node_metadata.o [ 132s] [138/231] CXX obj/src/liblwnode.node_config.o [ 132s] [139/231] CXX obj/src/liblwnode.node_symbols.o [ 132s] [140/231] CXX obj/src/liblwnode.node_sockaddr.o [ 133s] [141/231] CXX obj/src/liblwnode.node_os.o [ 133s] [142/231] CXX obj/src/liblwnode.node_report_utils.o [ 133s] [143/231] CXX obj/src/liblwnode.node_stat_watcher.o [ 133s] [144/231] CXX obj/src/liblwnode.node_credentials.o [ 134s] [145/231] CXX obj/src/liblwnode.node_binding.o [ 134s] [146/231] CXX obj/src/liblwnode.js_stream.o [ 134s] [147/231] CXX obj/src/liblwnode.histogram.o [ 134s] [148/231] CXX obj/src/liblwnode.pipe_wrap.o [ 134s] [149/231] CXX obj/src/liblwnode.node_main_instance.o [ 134s] [150/231] CXX obj/src/liblwnode.node_dir.o [ 134s] [151/231] CXX obj/src/liblwnode.node_env_var.o [ 134s] [152/231] CXX obj/src/liblwnode.node_v8.o [ 135s] [153/231] CXX obj/src/liblwnode.node_report_module.o [ 135s] [154/231] CXX obj/src/liblwnode.node_serdes.o [ 135s] [155/231] CXX obj/src/liblwnode.js_udp_wrap.o [ 135s] [156/231] CXX obj/src/liblwnode.node_perf.o [ 135s] [157/231] CXX obj/src/liblwnode.node_report.o [ 135s] [158/231] CXX obj/src/liblwnode.module_wrap.o [ 135s] [159/231] CXX obj/src/liblwnode.handle_wrap.o [ 135s] [160/231] CXX obj/src/liblwnode.fs_event_wrap.o [ 135s] [161/231] CXX obj/src/liblwnode.heap_utils.o [ 136s] [162/231] CXX obj/src/liblwnode.node.o [ 136s] [163/231] CXX obj/src/liblwnode.node_api.o [ 136s] [164/231] CXX obj/src/liblwnode.node_errors.o [ 136s] [165/231] CXX obj/src/liblwnode.connection_wrap.o [ 136s] [166/231] CXX obj/src/liblwnode.node_platform.o [ 137s] [167/231] CXX obj/src/liblwnode.node_process_events.o [ 137s] [168/231] CXX obj/src/liblwnode.connect_wrap.o [ 137s] [169/231] CXX obj/src/api/liblwnode.utils.o [ 137s] [170/231] CXX obj/src/liblwnode.node_zlib.o [ 137s] [171/231] CXX obj/src/liblwnode.node_postmortem_metadata.o [ 137s] [172/231] CXX obj/src/liblwnode.node_task_queue.o [ 137s] [173/231] CXX obj/src/liblwnode.node_types.o [ 138s] [174/231] CXX obj/src/liblwnode.js_native_api_v8.o [ 138s] [175/231] CXX obj/src/liblwnode.node_util.o [ 138s] [176/231] CXX obj/src/liblwnode.debug_utils.o [ 138s] [177/231] CXX obj/src/liblwnode.node_buffer.o [ 138s] [178/231] CXX obj/src/liblwnode.node_contextify.o [ 139s] [179/231] CXX obj/src/liblwnode.node_http_parser.o [ 139s] [180/231] CXX obj/src/liblwnode.node_process_object.o [ 139s] [181/231] CXX obj/src/api/liblwnode.exceptions.o [ 139s] [182/231] CXX obj/src/liblwnode.node_process_methods.o [ 139s] [183/231] CXX obj/src/api/liblwnode.hooks.o [ 139s] [184/231] CXX obj/src/api/liblwnode.encoding.o [ 140s] [185/231] CXX obj/src/liblwnode.node_trace_events.o [ 140s] [186/231] CXX obj/src/liblwnode.signal_wrap.o [ 140s] [187/231] CXX obj/src/liblwnode.node_constants.o [ 140s] [188/231] CXX obj/src/liblwnode.node_native_module.o [ 140s] [189/231] CXX obj/src/api/liblwnode.async_resource.o [ 140s] [190/231] CXX obj/src/large_pages/liblwnode.node_large_page.o [ 140s] [191/231] CXX obj/src/liblwnode.process_wrap.o [ 140s] [192/231] CXX obj/src/liblwnode.node_watchdog.o [ 140s] [193/231] CXX obj/src/api/liblwnode.callback.o [ 140s] [194/231] CXX obj/src/tracing/liblwnode.node_trace_buffer.o [ 140s] [195/231] CXX obj/src/liblwnode.node_worker.o [ 141s] [196/231] CXX obj/src/liblwnode.string_decoder.o [ 141s] [197/231] CXX obj/src/liblwnode.node_native_module_env.o [ 141s] [198/231] CXX obj/src/liblwnode.timers.o [ 141s] [199/231] CXX obj/src/liblwnode.node_url.o [ 141s] [200/231] CXX obj/src/tracing/liblwnode.traced_value.o [ 141s] [201/231] CXX obj/src/tracing/liblwnode.trace_event.o [ 141s] [202/231] CXX obj/src/liblwnode.string_bytes.o [ 141s] [203/231] CXX obj/src/tracing/liblwnode.node_trace_writer.o [ 141s] [204/231] CXX obj/src/liblwnode.node_crypto_clienthello.o [ 142s] [205/231] CXX obj/src/liblwnode.node_snapshot_stub.o [ 142s] [206/231] CXX obj/src/liblwnode.node_code_cache_stub.o [ 142s] [207/231] CXX obj/src/liblwnode.spawn_sync.o [ 142s] [208/231] CXX obj/src/liblwnode.stream_wrap.o [ 142s] [209/231] CXX obj/src/api/liblwnode.environment.o [ 142s] [210/231] CXX obj/src/liblwnode.node_file.o [ 142s] [211/231] CXX obj/src/liblwnode.stream_pipe.o [ 142s] [212/231] CXX obj/src/liblwnode.tcp_wrap.o [ 143s] [213/231] CXX obj/src/liblwnode.uv.o [ 143s] [214/231] CXX obj/src/liblwnode.tty_wrap.o [ 143s] [215/231] CXX obj/src/tracing/liblwnode.agent.o [ 143s] [216/231] CXX obj/gen/liblwnode.node_javascript.o [ 143s] [217/231] CXX obj/src/liblwnode.stream_base.o [ 143s] [218/231] CXX obj/src/liblwnode.node_options.o [ 143s] [219/231] CXX obj/src/liblwnode.node_wasi.o [ 143s] [220/231] CXX obj/src/liblwnode.util.o [ 144s] [221/231] CXX obj/src/liblwnode.node_crypto_bio.o [ 144s] [222/231] CXX obj/src/liblwnode.node_messaging.o [ 144s] [223/231] CXX obj/src/liblwnode.async_wrap.o [ 144s] [224/231] CXX obj/src/liblwnode.udp_wrap.o [ 144s] [225/231] CXX obj/src/liblwnode.cares_wrap.o [ 145s] [226/231] CXX obj/src/liblwnode.node_crypto_common.o [ 146s] [227/231] CXX obj/src/liblwnode.tls_wrap.o [ 151s] [228/231] CXX obj/src/liblwnode.env.o [ 153s] [229/231] CXX obj/src/liblwnode.node_crypto.o [ 157s] [230/231] CXX obj/src/liblwnode.node_http2.o [ 157s] ../../../src/node_http2.cc: In function 'void node::http2::Initialize(v8::Local, v8::Local, v8::Local, void*)': [ 157s] ../../../src/node_http2.cc:2953:6: note: variable tracking size limit exceeded with '-fvar-tracking-assignments', retrying without [ 157s] 2953 | void Initialize(Local target, [ 157s] | ^~~~~~~~~~ [ 161s] [231/231] SOLINK lib/liblwnode.so.83 [ 161s] + ./configure --tizen --without-npm --without-inspector --without-node-code-cache --without-node-snapshot --with-intl none --shared-zlib --shared-cares --shared-openssl --shared-nghttp2 --enable-reload-script --enable-external-builtin-script --dest-os linux --dest-cpu x32 --ninja --without-bundled-v8 --engine escargot [ 161s] Package openssl was not found in the pkg-config search path. [ 161s] Perhaps you should add the directory containing `openssl.pc' [ 161s] to the PKG_CONFIG_PATH environment variable [ 161s] No package 'openssl' found [ 162s] Node.js configure: Found Python 3.9.1... [ 162s] WARNING: No acceptable ld.gold linker found! [ 162s] LWNode.js options: [-Dbuild_mode=release -Dlwnode=true -Dlwnode_jsengine_path=lwnode/code/escargotshim -Denable_external_builtin_script=true -Denable_reload_script=true -Dtarget_os=tizen -Dprofile=common -Descargot_lib_type=shared_lib -Descargot_threading=0 -Dnode_core_target_name=lwnode -Dnode_lib_target_name=liblwnode] [ 162s] WARNING: warnings were emitted in the configure phase [ 162s] INFO: configure completed successfully [ 162s] + ninja -C out/tizen/Release lwnode [ 162s] ninja: Entering directory `out/tizen/Release' [ 162s] [1/230] CC obj/deps/uvwasi/src/uvwasi.wasi_rights.o [ 162s] [2/230] CC obj/deps/llhttp/src/llhttp.http.o [ 162s] [3/230] CC obj/deps/brotli/c/common/brotli.transform.o [ 162s] [4/230] CC obj/deps/brotli/c/dec/brotli.bit_reader.o [ 162s] [5/230] CC obj/deps/brotli/c/common/brotli.dictionary.o [ 162s] [6/230] CC obj/deps/brotli/c/common/brotli.platform.o [ 162s] [7/230] CC obj/deps/uv/src/unix/libuv.getaddrinfo.o [ 162s] [8/230] CC obj/deps/uv/src/unix/libuv.getnameinfo.o [ 162s] [9/230] CC obj/deps/uv/src/unix/libuv.loop.o [ 163s] [10/230] CC obj/deps/uv/src/unix/libuv.poll.o [ 163s] [11/230] CC obj/deps/uvwasi/src/uvwasi.wasi_serdes.o [ 163s] [12/230] CC obj/deps/uv/src/unix/libuv.loop-watcher.o [ 163s] [13/230] CC obj/deps/uv/src/unix/libuv.pipe.o [ 163s] [14/230] CC obj/deps/uv/src/unix/libuv.process.o [ 163s] [15/230] CC obj/deps/uv/src/unix/libuv.random-devurandom.o [ 163s] [16/230] CC obj/deps/uv/src/unix/libuv.tcp.o [ 163s] [17/230] CC obj/deps/uv/src/unix/libuv.thread.o [ 163s] [18/230] CC obj/deps/uv/src/unix/libuv.procfs-exepath.o [ 163s] [19/230] CC obj/deps/uv/src/unix/libuv.random-getrandom.o [ 163s] [20/230] CC obj/deps/uv/src/unix/libuv.random-sysctl-linux.o [ 163s] [21/230] CC obj/deps/uvwasi/src/uvwasi.clocks.o [ 163s] [22/230] CC obj/deps/uvwasi/src/uvwasi.uv_mapping.o [ 163s] [23/230] CC obj/deps/brotli/c/common/brotli.constants.o [ 163s] [24/230] CC obj/deps/brotli/c/common/brotli.context.o [ 163s] [25/230] CC obj/deps/llhttp/src/llhttp.api.o [ 163s] [26/230] CC obj/deps/uvwasi/src/uvwasi.poll_oneoff.o [ 163s] [27/230] CC obj/deps/histogram/src/histogram.hdr_histogram.o [ 163s] [28/230] CC obj/deps/uv/src/unix/libuv.fs.o [ 163s] ../../../deps/uv/src/unix/fs.c: In function 'uv__fs_sendfile': [ 163s] ../../../deps/uv/src/unix/fs.c:909:57: warning: passing argument 4 of 'uv__fs_copy_file_range' from incompatible pointer type [-Wincompatible-pointer-types] [ 163s] 909 | r = uv__fs_copy_file_range(in_fd, NULL, out_fd, &off, req->bufsml[0].len, 0); [ 163s] | ^~~~ [ 163s] | | [ 163s] | off_t * {aka long long int *} [ 163s] In file included from ../../../deps/uv/src/unix/internal.h:41, [ 163s] from ../../../deps/uv/src/unix/fs.c:30: [ 163s] ../../../deps/uv/src/unix/linux-syscalls.h:71:33: note: expected 'ssize_t *' {aka 'int *'} but argument is of type 'off_t *' {aka 'long long int *'} [ 163s] 71 | ssize_t* off_out, [ 163s] | ~~~~~~~~~^~~~~~~ [ 163s] ../../../deps/uv/src/unix/fs.c: In function 'uv__fs_copyfile': [ 163s] ../../../deps/uv/src/unix/fs.c:1223:20: warning: comparison of integer expressions of different signedness: '__fsword_t' {aka 'int'} and 'unsigned int' [-Wsign-compare] [ 163s] 1223 | if (s.f_type != /* CIFS */ 0xFF534D42u) [ 163s] | ^~ [ 163s] [29/230] CC obj/deps/uv/src/unix/libuv.stream.o [ 163s] [30/230] CC obj/deps/uv/src/unix/libuv.signal.o [ 163s] [31/230] CC obj/deps/uv/src/unix/libuv.tty.o [ 163s] [32/230] CC obj/deps/uv/src/unix/libuv.udp.o [ 163s] [33/230] CC obj/deps/uvwasi/src/uvwasi.fd_table.o [ 163s] [34/230] CC obj/deps/uvwasi/src/uvwasi.path_resolver.o [ 163s] [35/230] CXX obj/lwnode/code/escargotshim/src/api/utils/escargotshim.debug.o [ 163s] [36/230] CXX obj/lwnode/code/escargotshim/src/api/utils/escargotshim.string-util.o [ 163s] [37/230] CC obj/deps/uv/src/libuv.fs-poll.o [ 163s] [38/230] AR obj/deps/histogram/libhistogram.a [ 163s] [39/230] CC obj/deps/uvwasi/src/uvwasi.uvwasi.o [ 163s] [40/230] AR obj/deps/uvwasi/libuvwasi.a [ 163s] [41/230] CC obj/deps/uv/src/libuv.strscpy.o [ 163s] [42/230] CXX obj/lwnode/code/escargotshim/src/api/utils/logger/escargotshim.flags.o [ 163s] [43/230] CC obj/deps/uv/src/libuv.random.o [ 164s] [44/230] CC obj/deps/uv/src/libuv.uv-data-getter-setters.o [ 164s] [45/230] CC obj/deps/uv/src/libuv.version.o [ 164s] [46/230] CC obj/deps/uv/src/libuv.timer.o [ 164s] [47/230] CC obj/deps/uv/src/unix/libuv.async.o [ 164s] [48/230] CC obj/deps/uv/src/libuv.threadpool.o [ 164s] [49/230] CC obj/deps/llhttp/src/llhttp.llhttp.o [ 164s] ../../../deps/llhttp/src/llhttp.c: In function 'llhttp__internal__run': [ 164s] ../../../deps/llhttp/src/llhttp.c:8691:13: warning: unused variable 'avail' [-Wunused-variable] [ 164s] 8691 | int avail; [ 164s] | ^~~~~ [ 164s] ../../../deps/llhttp/src/llhttp.c:8869:13: warning: unused variable 'avail' [-Wunused-variable] [ 164s] 8869 | int avail; [ 164s] | ^~~~~ [ 164s] ../../../deps/llhttp/src/llhttp.c:9619:13: warning: unused variable 'avail' [-Wunused-variable] [ 164s] 9619 | int avail; [ 164s] | ^~~~~ [ 164s] [50/230] CC obj/deps/uv/src/libuv.inet.o [ 164s] [51/230] CC obj/deps/uv/src/unix/libuv.dl.o [ 164s] [52/230] AR obj/deps/llhttp/libllhttp.a [ 164s] [53/230] CXX obj/lwnode/code/escargotshim/src/libplatform/tracing/escargotshim.trace-object.o [ 164s] [54/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.es-v8-helper.o [ 164s] [55/230] CC obj/deps/uv/src/libuv.uv-common.o [ 165s] [56/230] CXX obj/lwnode/code/escargotshim/src/init/escargotshim.v8.o [ 165s] [57/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.arraybuffer-allocator.o [ 165s] [58/230] CC obj/deps/uv/src/unix/libuv.core.o [ 165s] [59/230] CXX obj/lwnode/code/escargotshim/src/libplatform/tracing/escargotshim.trace-config.o [ 165s] [60/230] CXX obj/lwnode/code/escargotshim/src/libplatform/tracing/escargotshim.trace-buffer.o [ 165s] [61/230] CC obj/lwnode/code/escargotshim/deps/minizip/minizip.ioapi.o [ 165s] [62/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.arraybuffer-deleter.o [ 165s] [63/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api-additionals.o [ 165s] [64/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api.o [ 166s] [65/230] CC obj/deps/uv/src/unix/libuv.proctitle.o [ 166s] [66/230] CC obj/deps/uv/src/libuv.idna.o [ 166s] [67/230] CXX obj/lwnode/code/escargotshim/src/libplatform/tracing/escargotshim.tracing-controller.o [ 166s] [68/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.handle.o [ 166s] [69/230] CC obj/deps/uv/src/unix/libuv.linux-syscalls.o [ 166s] [70/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.function.o [ 166s] [71/230] CC obj/deps/uv/src/unix/libuv.linux-inotify.o [ 166s] [72/230] CXX obj/lwnode/code/escargotshim/src/libplatform/tracing/escargotshim.trace-writer.o [ 166s] [73/230] CC obj/deps/brotli/c/dec/brotli.huffman.o [ 166s] [74/230] CXX obj/lwnode/code/escargotshim/src/libplatform/escargotshim.default-platform.o [ 166s] [75/230] CC obj/deps/uv/src/unix/libuv.linux-core.o [ 166s] [76/230] CXX obj/lwnode/code/escargotshim/src/api/utils/escargotshim.gc.o [ 166s] [77/230] AR obj/deps/uv/libuv.a [ 166s] [78/230] CC obj/deps/brotli/c/dec/brotli.state.o [ 166s] [79/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api-template.o [ 167s] [80/230] CXX obj/lwnode/code/escargotshim/src/api/utils/escargotshim.smaps.o [ 167s] [81/230] CC obj/lwnode/code/escargotshim/deps/minizip/minizip.unzip.o [ 167s] [82/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api-scripts.o [ 167s] [83/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api-exception.o [ 167s] [84/230] CXX obj/lwnode/code/escargotshim/src/execution/escargotshim.v8threads.o [ 167s] [85/230] AR obj/lwnode/code/escargotshim/deps/minizip/libminizip.a [ 167s] [86/230] CXX obj/lwnode/code/escargotshim/src/lwnode/escargotshim.lwnode-gc-strategy.o [ 167s] [87/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.handlescope.o [ 167s] [88/230] CC obj/deps/brotli/c/enc/brotli.fast_log.o [ 167s] [89/230] CC obj/deps/brotli/c/enc/brotli.entropy_encode.o [ 167s] [90/230] CC obj/deps/brotli/c/enc/brotli.command.o [ 167s] [91/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api-handles.o [ 167s] [92/230] CC obj/deps/brotli/c/enc/brotli.dictionary_hash.o [ 167s] [93/230] CC obj/deps/brotli/c/enc/brotli.encoder_dict.o [ 167s] [94/230] CC obj/deps/brotli/c/enc/brotli.memory.o [ 167s] [95/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.extra-data.o [ 167s] [96/230] CC obj/deps/brotli/c/enc/brotli.literal_cost.o [ 167s] [97/230] CC obj/deps/brotli/c/enc/brotli.utf8_util.o [ 167s] [98/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.context.o [ 167s] [99/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.stack-trace.o [ 167s] [100/230] CC obj/deps/brotli/c/enc/brotli.bit_cost.o [ 168s] [101/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.object.o [ 168s] [102/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.global-handles.o [ 168s] [103/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.internal.o [ 168s] [104/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api-serialization.o [ 168s] [105/230] CXX obj/lwnode/code/escargotshim/src/lwnode/escargotshim.lwnode.o [ 168s] [106/230] CC obj/deps/brotli/c/enc/brotli.cluster.o [ 168s] [107/230] CC obj/deps/brotli/c/enc/brotli.histogram.o [ 168s] [108/230] CC obj/deps/brotli/c/enc/brotli.brotli_bit_stream.o [ 168s] [109/230] CC obj/deps/brotli/c/enc/brotli.static_dict.o [ 168s] [110/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api-debug.o [ 169s] [111/230] CXX obj/lwnode/code/escargotshim/src/lwnode/escargotshim.lwnode-loader.o [ 169s] [112/230] CC obj/deps/brotli/c/enc/brotli.backward_references_hq.o [ 169s] [113/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.isolate.o [ 169s] [114/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.es-helper.o [ 169s] [115/230] CXX obj/lwnode/code/escargotshim/src/api/escargotshim.engine.o [ 169s] [116/230] ACTION liblwnode: node_js2c_16c8176d30d05044f5cbb72c692d4563 [ 169s] [117/230] STAMP obj/liblwnode.actions_rules_copies.stamp [ 169s] [118/230] CC obj/deps/brotli/c/enc/brotli.block_splitter.o [ 169s] [119/230] CC obj/deps/brotli/c/enc/brotli.metablock.o [ 169s] [120/230] CC obj/deps/brotli/c/dec/brotli.decode.o [ 170s] [121/230] CC obj/deps/brotli/c/enc/brotli.compress_fragment_two_pass.o [ 170s] [122/230] CXX obj/src/liblwnode.json_utils.o [ 171s] [123/230] CC obj/deps/brotli/c/enc/brotli.encode.o [ 171s] [124/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api-data.o [ 171s] [125/230] CC obj/deps/brotli/c/enc/brotli.compress_fragment.o [ 172s] [126/230] CXX obj/src/liblwnode.node_config.o [ 173s] [127/230] CXX obj/lwnode/code/escargotshim/src/api/utils/logger/escargotshim.logger.o [ 174s] [128/230] CXX obj/src/liblwnode.node_util.o [ 174s] [129/230] CXX obj/src/liblwnode.pipe_wrap.o [ 174s] [130/230] CXX obj/src/liblwnode.node_v8.o [ 174s] [131/230] CXX obj/src/liblwnode.js_udp_wrap.o [ 174s] [132/230] CXX obj/src/liblwnode.node_i18n.o [ 174s] [133/230] STAMP obj/lwnode.compile_depends.stamp [ 174s] [134/230] CXX obj/src/liblwnode.node_report_module.o [ 174s] [135/230] CXX obj/src/liblwnode.node_binding.o [ 174s] [136/230] CC obj/deps/brotli/c/enc/brotli.backward_references.o [ 174s] [137/230] CXX obj/src/liblwnode.node_env_var.o [ 174s] [138/230] AR obj/deps/brotli/libbrotli.a [ 175s] [139/230] CXX obj/src/liblwnode.node_process_object.o [ 175s] [140/230] CXX obj/src/liblwnode.node_report_utils.o [ 175s] [141/230] CXX obj/src/liblwnode.js_stream.o [ 175s] [142/230] CXX obj/src/liblwnode.histogram.o [ 175s] [143/230] CXX obj/src/lwnode.node_main.o [ 175s] [144/230] CXX obj/src/lwnode.node_code_cache_stub.o [ 175s] [145/230] CXX obj/src/liblwnode.node_sockaddr.o [ 175s] [146/230] CXX obj/src/liblwnode.node_api.o [ 176s] [147/230] CXX obj/src/lwnode.node_snapshot_stub.o [ 176s] [148/230] CXX obj/src/liblwnode.node_process_methods.o [ 176s] [149/230] CXX obj/src/liblwnode.node_serdes.o [ 176s] [150/230] CXX obj/src/liblwnode.node_task_queue.o [ 176s] [151/230] CXX obj/src/liblwnode.node_dir.o [ 176s] [152/230] CXX obj/src/liblwnode.node_types.o [ 177s] [153/230] CXX obj/src/liblwnode.node_platform.o [ 177s] [154/230] CXX obj/src/liblwnode.heap_utils.o [ 177s] [155/230] CXX obj/src/liblwnode.module_wrap.o [ 178s] [156/230] CXX obj/src/liblwnode.node_http_parser.o [ 178s] [157/230] CXX obj/src/liblwnode.node_zlib.o [ 178s] [158/230] CXX obj/src/liblwnode.node_trace_events.o [ 178s] [159/230] CXX obj/src/liblwnode.node_postmortem_metadata.o [ 178s] [160/230] CXX obj/src/liblwnode.node_symbols.o [ 178s] [161/230] CXX obj/src/liblwnode.js_native_api_v8.o [ 178s] [162/230] CXX obj/src/liblwnode.node_metadata.o [ 179s] [163/230] CXX obj/src/liblwnode.node_main_instance.o [ 179s] [164/230] CXX obj/lwnode/code/escargotshim/src/escargotshim.api-environment.o [ 179s] [165/230] CXX obj/src/liblwnode.node_errors.o [ 179s] [166/230] AR obj/lwnode/code/escargotshim/libescargotshim.a [ 179s] [167/230] CXX obj/src/liblwnode.node_process_events.o [ 179s] [168/230] CXX obj/src/api/liblwnode.utils.o [ 180s] [169/230] CXX obj/src/liblwnode.node_report.o [ 180s] [170/230] CXX obj/src/liblwnode.node_worker.o [ 180s] [171/230] CXX obj/src/liblwnode.node_buffer.o [ 180s] [172/230] CXX obj/src/liblwnode.fs_event_wrap.o [ 180s] [173/230] CXX obj/src/liblwnode.node_os.o [ 180s] [174/230] CXX obj/src/liblwnode.node.o [ 180s] [175/230] CXX obj/src/liblwnode.handle_wrap.o [ 180s] [176/230] CXX obj/src/liblwnode.connect_wrap.o [ 180s] [177/230] CXX obj/src/liblwnode.node_credentials.o [ 181s] [178/230] CXX obj/src/liblwnode.connection_wrap.o [ 181s] [179/230] CXX obj/src/liblwnode.node_stat_watcher.o [ 181s] [180/230] CXX obj/src/liblwnode.node_url.o [ 181s] [181/230] CXX obj/src/liblwnode.node_native_module.o [ 181s] [182/230] CXX obj/src/liblwnode.debug_utils.o [ 182s] [183/230] CXX obj/src/api/liblwnode.exceptions.o [ 182s] [184/230] CXX obj/src/api/liblwnode.encoding.o [ 182s] [185/230] CXX obj/src/liblwnode.node_wasi.o [ 183s] [186/230] CXX obj/src/liblwnode.node_file.o [ 183s] [187/230] CXX obj/src/liblwnode.node_perf.o [ 183s] [188/230] CXX obj/src/api/liblwnode.async_resource.o [ 183s] [189/230] CXX obj/src/api/liblwnode.callback.o [ 183s] [190/230] CXX obj/src/large_pages/liblwnode.node_large_page.o [ 183s] [191/230] CXX obj/src/api/liblwnode.hooks.o [ 183s] [192/230] CXX obj/src/liblwnode.node_native_module_env.o [ 183s] [193/230] CXX obj/src/liblwnode.node_messaging.o [ 184s] [194/230] CXX obj/src/tracing/liblwnode.trace_event.o [ 184s] [195/230] CXX obj/src/tracing/liblwnode.traced_value.o [ 184s] [196/230] CXX obj/src/tracing/liblwnode.node_trace_buffer.o [ 184s] [197/230] CXX obj/src/tracing/liblwnode.node_trace_writer.o [ 184s] [198/230] CXX obj/src/liblwnode.signal_wrap.o [ 184s] [199/230] CXX obj/src/liblwnode.stream_pipe.o [ 184s] [200/230] CXX obj/src/liblwnode.node_contextify.o [ 184s] [201/230] CXX obj/src/liblwnode.timers.o [ 184s] [202/230] CXX obj/src/api/liblwnode.environment.o [ 185s] [203/230] CXX obj/src/liblwnode.node_crypto_clienthello.o [ 185s] [204/230] CXX obj/src/liblwnode.process_wrap.o [ 185s] [205/230] CXX obj/src/liblwnode.spawn_sync.o [ 185s] [206/230] CXX obj/src/liblwnode.node_watchdog.o [ 185s] [207/230] CXX obj/src/liblwnode.string_decoder.o [ 185s] [208/230] CXX obj/src/liblwnode.string_bytes.o [ 185s] [209/230] CXX obj/src/liblwnode.tcp_wrap.o [ 185s] [210/230] CXX obj/src/liblwnode.node_constants.o [ 185s] [211/230] CXX obj/src/liblwnode.tty_wrap.o [ 185s] [212/230] CXX obj/src/liblwnode.stream_wrap.o [ 186s] [213/230] CXX obj/gen/liblwnode.node_javascript.o [ 186s] [214/230] CXX obj/src/liblwnode.stream_base.o [ 186s] [215/230] CXX obj/src/tracing/liblwnode.agent.o [ 186s] [216/230] CXX obj/src/liblwnode.uv.o [ 187s] [217/230] CXX obj/src/liblwnode.node_crypto_bio.o [ 187s] [218/230] CXX obj/src/liblwnode.async_wrap.o [ 187s] [219/230] CXX obj/src/liblwnode.udp_wrap.o [ 187s] [220/230] CXX obj/src/liblwnode.util.o [ 188s] [221/230] CXX obj/src/liblwnode.cares_wrap.o [ 188s] [222/230] CXX obj/src/liblwnode.node_crypto_common.o [ 189s] [223/230] CXX obj/src/liblwnode.tls_wrap.o [ 190s] [224/230] CXX obj/src/liblwnode.node_options.o [ 194s] [225/230] CXX obj/src/liblwnode.env.o [ 196s] [226/230] CXX obj/src/liblwnode.node_crypto.o [ 198s] [227/230] CXX obj/src/liblwnode.node_http2.o [ 198s] ../../../src/node_http2.cc: In function 'void node::http2::Initialize(v8::Local, v8::Local, v8::Local, void*)': [ 198s] ../../../src/node_http2.cc:2953:6: note: variable tracking size limit exceeded with '-fvar-tracking-assignments', retrying without [ 198s] 2953 | void Initialize(Local target, [ 198s] | ^~~~~~~~~~ [ 198s] [228/230] AR obj/liblwnode.a [ 198s] [229/230] STAMP obj/lwnode.actions_depends.stamp [ 202s] [230/230] LINK lwnode [ 202s] + exit 0 [ 202s] Executing(%install): /bin/sh -e /var/tmp/rpm-tmp.wjt93x [ 202s] + umask 022 [ 202s] + cd /home/abuild/rpmbuild/BUILD [ 202s] + cd lwnode-1.0.0 [ 202s] + LANG=C [ 202s] + export LANG [ 202s] + unset DISPLAY [ 202s] + rm -rf /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386 [ 202s] + mkdir -p /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386 [ 202s] + rm -rf /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386 [ 202s] + mkdir -p /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/bin [ 202s] + mkdir -p /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/lib [ 202s] + rm -f 'out/tizen/Release/lib/*.tmp' out/tizen/Release/lib/liblwnode.so.83.TOC [ 202s] + cp out/tizen/Release/gen/escargot/libescargot.so /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/lib [ 202s] + cp out/tizen/Release/lib/liblwnode.so.83 /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/lib [ 202s] + strip -v -g out/tizen/Release/lwnode [ 202s] copy from `out/tizen/Release/lwnode' [elf32-i386] to `out/tizen/Release/stlW38uC' [elf32-i386] [ 202s] + cp out/tizen/Release/lwnode /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/bin [ 202s] + cp out/tizen/Release/lwnode.dat /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/bin [ 202s] + /usr/lib/rpm/find-debuginfo.sh -j8 --build-id-seed 1.0.0-1 --unique-debug-src-base lwnode-1.0.0-1.i386 -S debugsourcefiles.list /home/abuild/rpmbuild/BUILD/lwnode-1.0.0 [ 202s] WARNING: /usr/bin/lwnode is already stripped! [ 202s] 1+0 records in [ 202s] 1+0 records out [ 202s] 7 bytes (7 B) copied, 1.1016e-05 s, 635 kB/s [ 202s] 1+0 records in [ 202s] 1+0 records out [ 202s] 7 bytes (7 B) copied, 1.5304e-05 s, 457 kB/s [ 202s] extracting debug info from /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/lib/libescargot.so [ 202s] extracting debug info from /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/lib/liblwnode.so.83 [ 205s] 0+0 records in [ 205s] 0+0 records out [ 205s] 0 bytes (0 B) copied, 2.1331e-05 s, 0.0 kB/s [ 205s] 0+0 records in [ 205s] 0+0 records out [ 205s] 0 bytes (0 B) copied, 1.1658e-05 s, 0.0 kB/s [ 205s] 30054 blocks [ 206s] + cp /usr/lib/rpm/debug.manifest /home/abuild/rpmbuild/BUILD/lwnode-1.0.0/lwnode-debuginfo.manifest [ 206s] + cp /usr/lib/rpm/debug.manifest /home/abuild/rpmbuild/BUILD/lwnode-1.0.0/lwnode-debugsource.manifest [ 206s] + /usr/lib/rpm/check-buildroot [ 206s] + /usr/lib/rpm//brp-compress [ 206s] + /usr/lib/rpm/brp-strip-static-archive /bin/strip [ 217s] + /usr/lib/rpm/brp-python-hardlink [ 217s] + /usr/lib/rpm/tizen/find-docs.sh /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386 [ 217s] + /usr/lib/rpm/tizen/find-isufiles.sh /home/abuild/rpmbuild/BUILD/lwnode-1.0.0/isu.list [ 217s] ~/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386 ~/rpmbuild/BUILD/lwnode-1.0.0 [ 217s] ~/rpmbuild/BUILD/lwnode-1.0.0 [ 217s] Processing files: lwnode-1.0.0-1.i686 [ 217s] Aptk: PG1hbmlmZXN0PgogICAgICAgIDxyZXF1ZXN0PgogICAgICAgICAgICAgICAgPGRv [ 217s] bWFpbiBuYW1lPSJfIi8+CiAgICAgICAgPC9yZXF1ZXN0Pgo8L21hbmlmZXN0Pgo= [ 217s] [ 217s] Executing(%license): /bin/sh -e /var/tmp/rpm-tmp.RHEyFy [ 217s] + umask 022 [ 217s] + cd /home/abuild/rpmbuild/BUILD [ 217s] + cd lwnode-1.0.0 [ 217s] + LICENSEDIR=/home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + export LC_ALL=C [ 217s] + LC_ALL=C [ 217s] + export LICENSEDIR [ 217s] + /bin/mkdir -p /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE.Apache-2.0 /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE.NodeJS /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE.MIT /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE.BSD-2-Clause /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE.BSD-3-Clause /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE.BOEHM-GC /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE.ICU /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE.LGPL-2.1+ /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + cp -pr LICENSE.Zlib /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386/usr/share/licenses/lwnode [ 217s] + exit 0 [ 217s] Provides: libescargot.so liblwnode.so.83 lwnode = 1.0.0-1 lwnode(x86-32) = 1.0.0-1 [ 217s] Requires(interp): /bin/sh /bin/sh [ 217s] Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 [ 217s] Requires(post): /bin/sh [ 217s] Requires(postun): /bin/sh [ 217s] Requires: ld-linux.so.2 ld-linux.so.2(GLIBC_2.1) ld-linux.so.2(GLIBC_2.3) libc.so.6 libc.so.6(GLIBC_2.0) libc.so.6(GLIBC_2.1) libc.so.6(GLIBC_2.1.2) libc.so.6(GLIBC_2.1.3) libc.so.6(GLIBC_2.10) libc.so.6(GLIBC_2.11) libc.so.6(GLIBC_2.17) libc.so.6(GLIBC_2.2) libc.so.6(GLIBC_2.2.4) libc.so.6(GLIBC_2.28) libc.so.6(GLIBC_2.3) libc.so.6(GLIBC_2.3.2) libc.so.6(GLIBC_2.3.3) libc.so.6(GLIBC_2.3.4) libc.so.6(GLIBC_2.4) libc.so.6(GLIBC_2.6) libc.so.6(GLIBC_2.7) libc.so.6(GLIBC_2.9) libcares.so.2 libcrypto.so.1.1 libcrypto.so.1.1(OPENSSL_1_1_0) libcrypto.so.1.1(OPENSSL_1_1_1) libcrypto.so.1.1(OPENSSL_1_1_1b) libdl.so.2 libdl.so.2(GLIBC_2.0) libdl.so.2(GLIBC_2.1) libdlog.so.0 libescargot.so libgcc_s.so.1 libgcc_s.so.1(GCC_3.0) libgcc_s.so.1(GLIBC_2.0) libm.so.6 libm.so.6(GLIBC_2.0) libm.so.6(GLIBC_2.1) libm.so.6(GLIBC_2.29) libnghttp2.so.14 libpthread.so.0 libpthread.so.0(GLIBC_2.0) libpthread.so.0(GLIBC_2.1) libpthread.so.0(GLIBC_2.2) libpthread.so.0(GLIBC_2.2.3) libpthread.so.0(GLIBC_2.3.2) libpthread.so.0(GLIBC_2.3.3) libssl.so.1.1 libssl.so.1.1(OPENSSL_1_1_0) libssl.so.1.1(OPENSSL_1_1_1) libstdc++.so.6 libstdc++.so.6(CXXABI_1.3) libstdc++.so.6(CXXABI_1.3.5) libstdc++.so.6(CXXABI_1.3.8) libstdc++.so.6(CXXABI_1.3.9) libstdc++.so.6(GLIBCXX_3.4) libstdc++.so.6(GLIBCXX_3.4.11) libstdc++.so.6(GLIBCXX_3.4.14) libstdc++.so.6(GLIBCXX_3.4.15) libstdc++.so.6(GLIBCXX_3.4.18) libstdc++.so.6(GLIBCXX_3.4.19) libstdc++.so.6(GLIBCXX_3.4.20) libstdc++.so.6(GLIBCXX_3.4.21) libstdc++.so.6(GLIBCXX_3.4.22) libstdc++.so.6(GLIBCXX_3.4.26) libstdc++.so.6(GLIBCXX_3.4.9) libz.so.1 [ 217s] Processing files: lwnode-debugsource-1.0.0-1.i686 [ 217s] Aptk: PG1hbmlmZXN0PgogPHJlcXVlc3Q+CiAgICA8ZG9tYWluIG5hbWU9Il8iLz4KIDwv [ 217s] cmVxdWVzdD4KPC9tYW5pZmVzdD4K [ 217s] [ 219s] Provides: lwnode-debugsource = 1.0.0-1 lwnode-debugsource(x86-32) = 1.0.0-1 [ 219s] Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 [ 219s] Processing files: lwnode-devel-1.0.0-1.i686 [ 219s] Aptk: PG1hbmlmZXN0PgogICAgICAgIDxyZXF1ZXN0PgogICAgICAgICAgICAgICAgPGRv [ 219s] bWFpbiBuYW1lPSJfIi8+CiAgICAgICAgPC9yZXF1ZXN0Pgo8L21hbmlmZXN0Pgo= [ 219s] [ 219s] Provides: lwnode-devel = 1.0.0-1 lwnode-devel(x86-32) = 1.0.0-1 [ 219s] Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 [ 219s] Requires: ld-linux.so.2 ld-linux.so.2(GLIBC_2.3) libc.so.6 libc.so.6(GLIBC_2.0) libc.so.6(GLIBC_2.1) libc.so.6(GLIBC_2.1.2) libc.so.6(GLIBC_2.1.3) libc.so.6(GLIBC_2.10) libc.so.6(GLIBC_2.17) libc.so.6(GLIBC_2.2) libc.so.6(GLIBC_2.2.4) libc.so.6(GLIBC_2.28) libc.so.6(GLIBC_2.3) libc.so.6(GLIBC_2.3.2) libc.so.6(GLIBC_2.3.3) libc.so.6(GLIBC_2.3.4) libc.so.6(GLIBC_2.4) libc.so.6(GLIBC_2.6) libc.so.6(GLIBC_2.7) libc.so.6(GLIBC_2.9) libcares.so.2 libcrypto.so.1.1 libcrypto.so.1.1(OPENSSL_1_1_0) libcrypto.so.1.1(OPENSSL_1_1_1) libcrypto.so.1.1(OPENSSL_1_1_1b) libdl.so.2 libdl.so.2(GLIBC_2.0) libdl.so.2(GLIBC_2.1) libescargot.so libgcc_s.so.1 libgcc_s.so.1(GCC_3.0) libgcc_s.so.1(GLIBC_2.0) libm.so.6 libm.so.6(GLIBC_2.0) libm.so.6(GLIBC_2.1) libm.so.6(GLIBC_2.29) libnghttp2.so.14 libpthread.so.0 libpthread.so.0(GLIBC_2.0) libpthread.so.0(GLIBC_2.1) libpthread.so.0(GLIBC_2.2) libpthread.so.0(GLIBC_2.3.2) libpthread.so.0(GLIBC_2.3.3) libssl.so.1.1 libssl.so.1.1(OPENSSL_1_1_0) libssl.so.1.1(OPENSSL_1_1_1) libstdc++.so.6 libstdc++.so.6(CXXABI_1.3) libstdc++.so.6(CXXABI_1.3.5) libstdc++.so.6(CXXABI_1.3.8) libstdc++.so.6(CXXABI_1.3.9) libstdc++.so.6(GLIBCXX_3.4) libstdc++.so.6(GLIBCXX_3.4.11) libstdc++.so.6(GLIBCXX_3.4.14) libstdc++.so.6(GLIBCXX_3.4.15) libstdc++.so.6(GLIBCXX_3.4.18) libstdc++.so.6(GLIBCXX_3.4.19) libstdc++.so.6(GLIBCXX_3.4.20) libstdc++.so.6(GLIBCXX_3.4.21) libstdc++.so.6(GLIBCXX_3.4.22) libstdc++.so.6(GLIBCXX_3.4.26) libstdc++.so.6(GLIBCXX_3.4.9) libz.so.1 [ 219s] Processing files: lwnode-debuginfo-1.0.0-1.i686 [ 219s] Aptk: PG1hbmlmZXN0PgogPHJlcXVlc3Q+CiAgICA8ZG9tYWluIG5hbWU9Il8iLz4KIDwv [ 219s] cmVxdWVzdD4KPC9tYW5pZmVzdD4K [ 219s] [ 220s] Provides: lwnode-debuginfo = 1.0.0-1 lwnode-debuginfo(x86-32) = 1.0.0-1 [ 220s] Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 [ 220s] Recommends: lwnode-debugsource(x86-32) = 1.0.0-1 [ 220s] Checking for unpackaged file(s): /usr/lib/rpm/check-files /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386 [ 224s] Wrote: /home/abuild/rpmbuild/SRPMS/lwnode-1.0.0-1.src.rpm [ 227s] XZ: Adjusted the number of threads from 32 to 21 to not exceed the memory usage limit of 3758096381 bytesWrote: /home/abuild/rpmbuild/RPMS/i686/lwnode-1.0.0-1.i686.rpm [ 233s] XZ: Adjusted the number of threads from 32 to 21 to not exceed the memory usage limit of 3758096381 bytesWrote: /home/abuild/rpmbuild/RPMS/i686/lwnode-debugsource-1.0.0-1.i686.rpm [ 236s] XZ: Adjusted the number of threads from 32 to 21 to not exceed the memory usage limit of 3758096381 bytesWrote: /home/abuild/rpmbuild/RPMS/i686/lwnode-devel-1.0.0-1.i686.rpm [ 250s] XZ: Adjusted the number of threads from 32 to 21 to not exceed the memory usage limit of 3758096381 bytesWrote: /home/abuild/rpmbuild/RPMS/i686/lwnode-debuginfo-1.0.0-1.i686.rpm [ 250s] Executing(%clean): /bin/sh -e /var/tmp/rpm-tmp.fgHaty [ 250s] + umask 022 [ 250s] + cd /home/abuild/rpmbuild/BUILD [ 250s] + cd lwnode-1.0.0 [ 250s] + rm -rf /home/abuild/rpmbuild/BUILDROOT/lwnode-1.0.0-1.i386 [ 250s] + rm -fr ./debugfiles.list ./debuglinks.list ./debugsourcefiles.list ./debugsources.list ./documentation.list ./elfbins.list [ 250s] + rm -fr ./lwnode-debuginfo.manifest ./lwnode-debugsource.manifest [ 250s] + exit 0 [ 250s] ... checking for files with abuild user/group [ 250s] ... creating baselibs [ 250s] [ 250s] ip-192-168-56-24 finished "build lwnode.spec" at Wed Apr 24 06:27:00 UTC 2024. [ 250s]