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<h1>SLEEF Vectorized Math Library<img src="sleeflogo3.png" alt="logo" width="25%" height="25%" style="float: right" /></h1>
<ul class="none" style="font-family: arial, sansserif; margin-top: 1.0cm;">
<li><a class="underlined" style="color:#a00000;" href="https://github.com/shibatch/sleef/releases">Download the latest version</a></li>
<li><a class="underlined" href="https://sleef.org/">Latest documentation on the web</a></li>
<li><a class="underlined" href="https://github.com/shibatch/sleef">GitHub repository</a></li>
<li><a class="underlined" href="https://sourceforge.net/p/sleef/mailman/">Mailing list</a></li>
<li><a class="underlined" href="https://sourceforge.net/p/sleef/discussion/general/">Discussion forum</a></li>
<li><a class="underlined" href="https://github.com/shibatch/sleef/wiki">Wiki pages</a></li>
</ul>
<h2>Table of contents</h2>
<ul class="none" style="font-family: arial, sansserif; padding-left: 0.5cm;">
<li><a class="underlined" href="index.xhtml">Introduction</a>
<ul class="disc">
<li><a href="index.xhtml#overview">Overview</a></li>
<li><a href="index.xhtml#environment">Supported environments</a></li>
<li><a href="index.xhtml#credit">Credit</a></li>
<li><a href="index.xhtml#partners">Partners</a></li>
<li><a href="index.xhtml#license">License</a></li>
<li><a href="index.xhtml#history">History</a></li>
<li><a href="index.xhtml#related">Related projects</a></li>
<li><a href="index.xhtml#publication">Publication</a></li>
</ul>
</li>
<li> </li>
<li><a class="underlined" href="compile.xhtml">Compiling and installing the library</a></li>
<li><a class="underlined" href="purec.xhtml">Math library reference</a></li>
<li><a class="underlined" href="dft.xhtml">DFT library reference</a></li>
<li><a class="underlined" href="misc.xhtml">Other tools included in the package</a></li>
<li><a class="underlined" href="benchmark.xhtml">Benchmark results</a></li>
<li><a class="underlined" href="additional.xhtml">Additional notes</a></li>
</ul>
<h2 id="overview">Overview</h2>
<p class="noindent">
SLEEF stands for SIMD Library for Evaluating Elementary
Functions. It implements vectorized versions of all C99 real
floating point math functions. It can utilize SIMD instructions that
are available on modern processors. SLEEF is designed to effciently
perform computation with SIMD instruction by reducing the use of
conditional branches and scatter/gather memory
access. Our <a class="underlined"
href="benchmark.xhtml">benchmarks</a> show that the performance of
SLEEF is comparable to that of the best commercial library. Unlike
vendor-tuned assembly-optimized libraries, SLEEF can be easily
ported to other architectures by writing a helper file, which is a
thin abstraction layer of SIMD intrinsics. SLEEF is also designed to
work with various operating systems and compilers. Link time
optimization can be used to reduce the overhead of calling
functions.
</p>
<p>
The library contains implementations of all C99 real FP math
functions in double precision and single precision. Different
accuracy of the results can be chosen for a subset of the elementary
functions; for this subset there are versions with up to 1 ULP error
(which is the maximum error, not the average) and even faster
versions with a few ULPs of error. For non-finite inputs and
outputs, the functions return correct results as specified in the
C99 standard. All the functions in the library are thoroughly tested
and confirmed that the evaluation error is within the designed
limit, by comparing the returned values against high-precision
evaluation using <a class="underlined"
href="http://www.mpfr.org/">the GNU MPFR Library</a>. Especially, we
carefully checked the error of the trigonometric functions with
arguments close to an integral multiple
of <i class="math">π</i>/2.
</p>
<p>
SLEEF also includes subroutines for discrete Fourier
transform(DFT). These subroutines are fully vectorized, heavily
unrolled, and parallelized in such a way that modern SIMD
instructions and multiple cores can be utilized for efficient
computation. It has an API similar to that of FFTW for easy
migration, and distributed under
<a class="underlined"
href="http://www.boost.org/users/license.html">BSL</a>, which is a
permissive open source license. The subroutines can utilize long
vectors up to 2048 bits. The helper files for abstracting SIMD
intrinsics are shared with SLEEF libm, and thus it is easy to port
DFT subroutines to other architectures. <a class="underlined"
href="https://github.com/shibatch/sleef/wiki/DFT-Performance">Preliminary
results of benchmark</a> are now available.
</p>
<h2 id="environment">Supported environments</h2>
<p class="noindent">
This library currently supports several SIMD architectures :
</p>
<div><br/></div>
<ul class="disc">
<li><a href="x86.xhtml">x86 - SSE2, SSE4.1, AVX, AVX2+FMA3, AVX512F</a></li>
<li><a href="aarch64.xhtml">AArch64 - Advanced SIMD, SVE</a></li>
<li><a href="aarch32.xhtml">AArch32 - NEON</a></li>
<li><a href="ppc64.xhtml">PowerPC64 - VSX</a></li>
<li><a href="s390x.xhtml">System/390 - ZVECTOR2</a></li>
</ul>
<div><br/></div>
<p>
In addition to the SIMD implementation, Pure C (scalar) version is
provided. The library provides dispatchers that automatically choose
the best subroutines for the computer the library is executed
on. The supported combinations of the architecture, operating system
and compiler are shown in Table 1.1.
</p>
<div style="margin-top: 1.0cm;"></div>
<table style="text-align:center;" align="center">
<tr align="center">
<td class="caption">Table 1.1: Environment support matrix</td>
</tr>
<tr align="center">
<td>
<table class="lt">
<tr>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
</tr>
<tr>
<td class="lt-br"></td>
<td class="lt-hl"></td>
<td class="lt-blr" align="center">GCC</td>
<td class="lt-br" align="center">Clang</td>
<td class="lt-br" align="center">Intel Compiler</td>
<td class="lt-b" align="center">MSVC</td>
</tr>
<tr>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
<td class="lt-hl"></td>
</tr>
<tr>
<td class="lt-r" align="left">x86_64, Linux</td>
<td></td>
<td class="lt-lr" align="center">Supported</td>
<td class="lt-r" align="center">Supported</td>
<td class="lt-r" align="center">Supported</td>
<td class="lt-" align="center">N/A</td>
</tr>
<tr>
<td class="lt-r" align="left">AArch64, Linux</td>
<td></td>
<td class="lt-lr" align="center">Supported</td>
<td class="lt-r" align="center">Supported</td>
<td class="lt-r" align="center">N/A</td>
<td class="lt-" align="center">N/A</td>
</tr>
<tr>
<td class="lt-r" align="left">AArch32, Linux</td>
<td></td>
<td class="lt-lr" align="center">Supported(*1)</td>
<td class="lt-r" align="center">Supported(*1)</td>
<td class="lt-r" align="center">N/A</td>
<td class="lt-" align="center">N/A</td>
</tr>
<tr>
<td class="lt-r" align="left">PowerPC, Linux</td>
<td></td>
<td class="lt-lr" align="center">Supported</td>
<td class="lt-r" align="center">Supported</td>
<td class="lt-r" align="center">N/A</td>
<td class="lt-" align="center">N/A</td>
</tr>
<tr>
<td class="lt-r" align="left">System/390, Linux</td>
<td></td>
<td class="lt-lr" align="center">Supported</td>
<td class="lt-r" align="center">Supported</td>
<td class="lt-r" align="center">N/A</td>
<td class="lt-" align="center">N/A</td>
</tr>
<tr>
<td class="lt-r" align="left">x86_64, FreeBSD</td>
<td></td>
<td class="lt-lr" align="center"></td>
<td class="lt-r" align="center">Supported</td>
<td class="lt-r" align="center">N/A</td>
<td class="lt-" align="center">N/A</td>
</tr>
<tr>
<td class="lt-r" align="left">x86_64, OS X</td>
<td></td>
<td class="lt-lr" align="center">Supported(*2)</td>
<td class="lt-r" align="center">Supported(*2)</td>
<td class="lt-r" align="center"></td>
<td class="lt-" align="center">N/A</td>
</tr>
<tr>
<td class="lt-br" align="left">x86_64, Windows</td>
<td class="lt-hl"></td>
<td class="lt-blr" align="center">Supported(Cygwin)(*3)</td>
<td class="lt-br" align="center">Supported(*3)</td>
<td class="lt-br" align="center"></td>
<td class="lt-b" align="center">Supported</td>
</tr>
</table>
</td>
</tr>
</table>
<div style="margin-top: 2.0cm;"></div>
<p>
The supported compiler versions are as follows.
</p>
<div><br/></div>
<ul class="disc">
<li>GCC : version 5 and later</li>
<li>Clang : version 6 and later</li>
<li>Intel Compiler : ICC version 17</li>
<li>MSVC : Visual Studio 2019</li>
</ul>
<div><br/></div>
<p>
*1 NEON has only single precision support. The computation results
are not in full accuracy because NEON is not IEEE 754-compliant.
</p>
<p>
*2 LTO is not supported.
</p>
<p>
*3 AVX functions are not supported for Cygwin, because AVX is not
supported by Cygwin ABI.
</p>
<div style="margin-top: 1.0cm;"></div>
<p>
All functions in the library are thread safe unless otherwise noted.
</p>
<h2 id="credit">Credit</h2>
<ul class="disc">
<li>The main developer is <a class="underlined"
href="https://github.com/shibatch">Naoki Shibata</a>
( <a href="mailto:shibatch@users.sourceforge.net">shibatch@users.sourceforge.net</a> )
at Nara Institute of Science and Technology.</li>
<li><a class="underlined"
href="https://github.com/fpetrogalli">Francesco Petrogalli</a>
at ARM Ltd. contributed the helper for AArch64 (helperadvsimd.h,
helpersve.h) and GNUABI interface of the library. He also worked on
migrating the build system to cmake, and reviewed the code, gave
precious comments and suggestions.</li>
<li><a class="underlined" href="https://github.com/hfinkel">Hal Finkel</a> at Argonne
Leadership Computing Facility is now working
on <a class="underlined"
href="https://reviews.llvm.org/D24951">importing and adapting SLEEF
as an LLVM runtime</a>. He also gave precious comments.</li>
<li><a class="underlined" href="https://github.com/diaena">Diana
Bite</a> at University of Manchester and <a class="underlined"
href="https://github.com/xoofx">Alexandre Mutel</a> at Unity
Technologies worked on migrating the build system to
cmake.</li>
<li><a class="underlined" href="https://github.com/blu">Martin
Krastev</a> at Chaos Group contributed faster implementation of fmin
and fmax functions.</li>
<li><a class="underlined" href="https://github.com/cdluminate">Mo Zhou
</a> is managing packages for <a class="underlined"
href="https://tracker.debian.org/pkg/sleef">Debian</a>
and <a class="underlined"
href="https://launchpad.net/~lumin0/+archive/ubuntu/sleef">Ubuntu
PPA</a>.</li>
</ul>
<h2 id="partners">Partner institutes and corporations</h2>
<table style="margin-top:1cm;">
<tr><td> </td></tr>
<tr>
<td style="text-align:center;"><a href="https://www.naist.jp/en/"><img src="naistlogo.svg"
height="135px" alt="NAIST logo"/></a></td>
<td style="vertical-align: top;">
<p class="noindent" style="margin-left:2em;">
Division of Information Science of Nara Institute
of Science and Technology participates through Naoki Shibata.
</p>
</td>
</tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr>
<td style="text-align:center;"><a href="https://www.arm.com/"><img src="https://sleef.org/Arm-logo-blue-pms313.svg" alt="ARM logo"
height="60px" /></a></td>
<td style="vertical-align: top;">
<p class="noindent" style="margin-left:2em;">
As the leading IP company in semiconductors
design, <a class="underlined" href="https://www.arm.com/">ARM</a>
participates through Francesco Petrogalli.
</p>
</td>
</tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr><td> </td></tr>
<tr>
<td style="text-align:center;"><a href="https://unity3d.com/"><img src="https://sleef.org/Unity_Technologies_logo.svg" alt="Unity Technologies logo"
height="92px" /></a></td>
<td style="vertical-align: top;">
<p class="noindent" style="margin-left:2em;">As the leading
company in developing a video game engine, <a class="underlined"
href="https://unity3d.com/">Unity Technologies</a> participates
through Alexandre Mutel.
</p>
</td>
</tr>
<tr><td> </td></tr>
</table>
<h2 id="license">License</h2>
<p class="noindent">
SLEEF is distributed under <a class="underlined"
href="http://www.boost.org/LICENSE_1_0.txt">Boost Software License
Version 1.0</a>.
</p>
<p class="noindent" style="margin-top: 0.8cm;">
<a class="nothing" href="https://opensource.org/">
<img src="osi_logo.png" width="90" height="103" alt="open source logo" style="position:relative; top:3px;"/>
</a>
Boost Software License is <a class="underlined"
href="https://opensource.org/licenses/BSL-1.0">OSI-certified</a>.
See <a class="underlined"
href="http://www.boost.org/users/license.html">this page</a> for more
information about Boost Software License.
</p>
<h2 id="history">History</h2>
<h4>3.5.1 (Released on Sep 15, 2020)</h4>
<ul class="disc">
<li>Fixed a bug in handling compiler options</li>
</ul>
<h4>3.5 (Released on Sep 1, 2020)</h4>
<ul class="disc">
<li>IBM System/390 support is added (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/291">#291</a>)</li>
<li>The library can be built with Clang on Windows (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/300">#300</a>)</li>
<li>Static libraries with LTO can be generated (PR
<a class="underlined" href="https://github.com/shibatch/sleef/pull/290">#290</a>)</li>
<li>Alternative division and sqrt methods can be chosen with AArch64 (PR
<a class="underlined" href="https://github.com/shibatch/sleef/pull/289">#289</a>)</li>
<li>Header files for inlining the whole SLEEF functions can be
generated (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/283">#283</a>)</li>
<li>IEEE remainder function is added (PR
<a class="underlined" href="https://github.com/shibatch/sleef/pull/271">#271</a>)</li>
<li>GCC-10 can now build SLEEF with SVE support (PR
<a class="underlined" href="https://github.com/shibatch/sleef/pull/310">#310</a>)</li>
</ul>
<h4>3.4.1 (Released on Oct 1, 2019)</h4>
<ul class="disc">
<li>Fixed accuracy problem with tan_u35, atan_u10, log2f_u35 and exp10f_u10 (PR
<a class="underlined" href="https://github.com/shibatch/sleef/pull/260">#260</a>,
<a class="underlined" href="https://github.com/shibatch/sleef/pull/265">#265</a>,
<a class="underlined" href="https://github.com/shibatch/sleef/pull/267">#267</a>)</li>
<li>SVE intrinsics that are not supported in newer ACLE are replaced (PR
<a class="underlined" href="https://github.com/shibatch/sleef/pull/268">#268</a>)</li>
<li>FMA4 detection problem is fixed (PR
<a class="underlined" href="https://github.com/shibatch/sleef/pull/262">#262</a>)</li>
<li>Compilation problem under Windows with MinGW is fixed (PR
<a class="underlined" href="https://github.com/shibatch/sleef/pull/266">#266</a>)</li>
</ul>
<h4>3.4 (Released on Apr 28, 2019)</h4>
<ul class="disc">
<li>Faster and low precision functions are
added (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/229">#229</a>)</li>
<li>Functions that return consistent results across platforms are
added (PR <a class="underlined" href="https://github.com/shibatch/sleef/pull/216">#216</a>,
<a class="underlined" href="https://github.com/shibatch/sleef/pull/224">#224</a>)</li>
<li>Many functions are now faster (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/239">#239</a>)</li>
<li>Quad precision math library(libsleefquad) is added
(PR <a class="underlined" href="https://github.com/shibatch/sleef/pull/235">#235</a>,
<a class="underlined" href="https://github.com/shibatch/sleef/pull/237">#237</a>,
<a class="underlined" href="https://github.com/shibatch/sleef/pull/240">#240</a>)</li>
<li>Testers are now faster (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/223">#223</a>)</li>
</ul>
<h4>3.3.1 (Released on Aug 21, 2018)</h4>
<ul class="disc">
<li>i386 build problem is fixed</li>
<li>FreeBSD support is added</li>
<li>Trigonometric functions now evaluate correctly with full FP
domain. (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/210">#210</a>)</li>
</ul>
<h4>3.3 (Released on July 6, 2018)</h4>
<ul class="disc">
<li>AArch64 SVE target support is added (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/180">#180</a>, <a class="underlined"
href="https://github.com/shibatch/sleef/pull/182">#182</a>)</li>
<li>DFT is now faster (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/186">#186</a>)</li>
<li>3.5-ULP hyperbolic functions are added (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/192">#192</a>)</li>
<li>PowerPC VSX target support is added (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/195">#195</a>)</li>
<li>Modified Payne-Hanek argument reduction is added to the
trigonometric functions in libsleef (PR <a class="underlined"
href="https://github.com/shibatch/sleef/pull/197">#197</a>)</li>
</ul>
<h4>3.2 (Released on Feb 26, 2018)</h4>
<ul class="disc">
<li>The whole build system of the project migrated from makefiles to
cmake. The makefile build system is now removed.</li>
<li>GNUABI version of the library with compatibility tests is
added.</li>
<li>Benchmarks that compare `libsleef` vs `SVML` on X86 Linux are
available in the project tree under src/libm-benchmarks
directory.</li>
<li>Extensive upstream testing via Travis CI and Appveyor</li>
<li>log2 is added.</li>
<li>The library can be compiled to an LLVM bitcode object</li>
<li>Added masked interface to the library to support AVX512F masked
vectorization.</li>
<li>Use native instructions if available for `sqrt`.</li>
<li>Removed `libm` dependency.</li>
<li>fmod(FP remainder), asin, acos, log, pow, log10, exp2, exp10 and
log1p functions are now faster.</li>
<li>Fixed a bug that was making the error of sinpi, cospi, sincospi,
and tgamma functions larger than the specifications on very rare
occasions.</li>
<li>Fixed a bug that was preventing the dispatcher from choosing the
FMA4 implementation.</li>
</ul>
<p class="noindent" style="margin-top: 1.0cm;">
See <a class="underlined"
href="https://github.com/shibatch/sleef/blob/master/CHANGELOG.md">Changelog</a>
for older changes.</p>
<h2 id="related">Related projects and links</h2>
<ul class="disc">
<li>SLEEF is adopted in <a class="underlined"
href="https://youtu.be/QkM6zEGFhDY?t=2328">the Burst Comipler in
Unity</a>, and also in
the <a class="underlined" href="https://youtu.be/WnJV6J-taIM?t=426">Android
version</a>.</li>
<li>SLEEF is now included in <a class="underlined"
href="https://developer.arm.com/documentation/101458/1930/Vector-math-routines/Vector-math-routines-in-Arm-C-C---Compiler">
Arm Compiler for HPC</a>.</li>
<li>Work for making SLEEF usable from the vectorizer
in <a class="underlined" href="http://llvm.org/">LLVM</a> is now
carried out(<a class="underlined"
href="https://reviews.llvm.org/D53927">D53927</a>, <a class="underlined"
href="https://reviews.llvm.org/D53928">D53928</a>,
<a class="underlined"
href="https://reviews.llvm.org/D54412">D54412</a>).</li>
<li>Mustafa A. Mohamad is developing <a class="underlined"
href="https://github.com/musm/SLEEF.jl">a pure Julia port of the
SLEEF math library</a>.</li>
<li>Hal Finkel has <a class="underlined"
href="https://github.com/hfinkel/sleef-bgq">adapted SLEEF to
PowerPC/QPX</a>. SLEEF is running on Mira IBM BG/Q supercomputer at
Argonne National Laboratory.</li>
<li><a class="underlined" href="https://github.com/gnzlbg">Gonzalo
Brito Gadeschi</a> is developing <a class="underlined"
href="https://docs.rs/crate/sleef-sys/">a Rust FFI bindings to
SLEEF</a>.</li>
<li>SLEEF is adopted in the following projects.
<ul class="circle">
<li><a href="http://www.menuetos.net/sc151.html">MenuetOS</a></li>
<li><a href="https://github.com/Beep6581/RawTherapee/tree/dev/rtengine">RawTherapee</a></li>
<li><a href="http://buraphakit.sourceforge.net/BASIC.shtml">John's ECMA-55 Minimal BASIC Compiler</a></li>
<li><a href="https://github.com/pocl/pocl/tree/master/lib/kernel/sleef">Portable Computing Language(pocl)</a></li>
<li><a href="https://github.com/pytorch/pytorch/pull/6725">PyTorch</a></li>
<li><a href="https://github.com/cdl-saarland/rv">The Region Vectorizer(Compiler Design Lab at Saarland University)</a></li>
<li><a href="https://agenium-scale.github.io/nsimd">Agenium Scale NSIMD</a></li>
<li><a href="https://code.jsoftware.com/wiki/System/ReleaseNotes/J902">J Language</a></li>
<li><a href="https://github.com/simd-everywhere/simde">SIMD Everywhere</a></li>
<li><a href="https://github.com/dnbaker/minocore">Minocore</a></li>
<li><a href="https://github.com/greatest-ape/OctaSine">OctaSine</a></li>
<li><a href="https://github.com/jackmott/simdeez">Simdeez</a></li>
</ul>
</li>
</ul>
<h2 id="publication">Publication</h2>
<ul class="disc">
<li>Naoki Shibata and Francesco Petrogalli : <B>SLEEF: A Portable Vectorized Library of C Standard Mathematical Functions,</B> <I>in IEEE Transactions on Parallel and Distributed Systems,</I> <a class="underlined" href="https://doi.org/10.1109/TPDS.2019.2960333">DOI:10.1109/TPDS.2019.2960333</a> (Dec. 2019). [<a class="underlined" href="https://arxiv.org/pdf/2001.09258">PDF</a>]</li>
<li>Francesco Petrogalli and Paul Walker : <B>LLVM and the automatic vectorization of loops invoking math routines: -fsimdmath</B>, <I>2018 IEEE/ACM 5th Workshop on the LLVM Compiler Infrastructure in HPC (LLVM-HPC), pp. 30-38.,</I> <a class="underlined" href="https://doi.org/10.1109/LLVM-HPC.2018.8639354">DOI:10.1109/LLVM-HPC.2018.8639354</a> (Nov. 2018). [<a class="underlined" href="https://sc18.supercomputing.org/proceedings/workshops/workshop_files/ws_llvmf106s2-file1.pdf">PDF</a>]</li>
</ul>
<p class="footer">
Copyright © <!--YEAR--> SLEEF Project.<br/>
SLEEF is open-source software and is distributed under the Boost Software License, Version 1.0.
</p>
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