Description: SLEEF implements vectorized C99 math functions. It utilizes vector extensions including SSE2, AVX, AVX2, AVX512F for x86, and Advanced SIMD for ARM processors.

# SLEEF Vectorized Math Library

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## Table of contents

- Introduction
  - Overview
  - Supported environments
  - Credit
  - Partners
  - License
  - History of this software
  - Related projects
- Compiling and installing the library
- Math library reference
- DFT library reference
- Other tools included in the package
- Benchmark results
- Additional notes

## Overview

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 of modern processors. SLEEF is designed to fully utilize SIMD computation by reducing the use of conditional branches and scatter/gather memory access. Our benchmarks show that the performance of SLEEF is comparable to that of the best commercial library. Unlike vendor-tuned libraries, SLEEF is portable : it 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.

The library contains subroutines for 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 versions with a few ulp error. Obviously, less accurate versions are faster. For non-finite inputs and outputs, the library should return the same results as libm as specified in the C99 standard. The library is rigorously tested if the evaluation error is within the designed limit. The library is tested against high-precision evaluation using the libmpfr library. Especially, we carefully checked the error of the trigonometric functions when the arguments are close to an integral multiple of π/2.

SLEEF also includes subroutines for discrete Fourier transform(DFT). These subroutines are fully vectorized, heavily unrolled, and parallelized so 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 BSL, which is a permissive open source license. The subroutines can utilize long vectors up to 2048 bits, and even longer vectors can be utilized by a small modification. The helper files for abstracting SIMD intrinsics are shared with SLEEF libm, and thus it is easy to port DFT subroutines to other architectures. Preliminary results of benchmark are now available.

## Supported environments

This library currently supports several SIMD architectures :

- x86 - SSE2, SSE4.1, AVX, FMA4, AVX2+FMA3, AVX512F
- AArch64 - Advanced SIMD, SVE
- AArch32 - NEON
- PowerPC64 - VSX

In addition to the SIMD implementation, Pure C (scalar) version is provided. For x86 architecture, the library provides dispatchers that automatically choose the best subroutines for the computer the library is run. The supported combinations of the architecture, operating system and compiler are shown in Table 1.1.

Table 1.1: Environment support matrix
|  | GCC | Clang | Intel Compiler | MSVC
x86 (64bit), Linux |  | Supported | Supported | Supported(*1) | N/A
x86 (32bit), Linux |  | Supported(*2) | Supported(*2) |  | N/A
AArch64, Linux |  | Supported | Supported | N/A | N/A
AArch32, Linux |  | Supported(*3) | Supported(*3) | N/A | N/A
PowerPC, Linux |  |  | Supported(*4) | N/A | N/A
x86 (64bit), OS X |  | Supported | Supported |  | N/A
x86 (64bit), Windows |  | Supported(Cygwin)(*5) | Supported(Cygwin)(*5) |  | Supported

The supported compiler versions are as follows.

- GCC : version 5 and later
- Clang : version 3.9 and later
- Intel Compiler : ICC version 17
- MSVC : Visual Studio 2017

*1 FMA4 is not supported by Intel Compiler.

*2 SSE2 is required to run the scalar functions on 32-bit x86 architecture. x87 is not supported.

*3 NEON has only single precision support. The computation results are not in full accuracy since NEON is not IEEE 754-compliant.

*4 Clang-5.0 and later are supported.

*5 AVX functions are not supported for Cygwin, since AVX is not supported by Cygwin ABI.

All functions in the library are thread safe unless otherwise noted.

## Credit

- The main developer is Naoki Shibata ( shibatch@users.sourceforge.net ) at Nara Institute of Science and Technology.
- Francesco Petrogalli 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.
- Hal Finkel at Argonne Leadership Computing Facility is now working on importing and adapting SLEEF as an LLVM runtime. He also gave precious comments.
- Diana Bite at University of Manchester and Alexandre Mutel at Unity Technologies worked on migrating the build system to cmake.
- Martin Krastev at Chaos Group contributed faster implementation of fmin and fmax functions.

## Partner institutes and corporations

| The Mobile Computing Lab at Division of Information Science of Nara Institute of Science and Technology participates through Naoki Shibata.
| As the leading IP company in semiconductors design, ARM participates through Francesco Petrogalli.
| As the leading company in developing a video game engine, Unity Technologies participates through Alexandre Mutel.

## License

SLEEF is distributed under Boost Software License Version 1.0.

Boost Software License is OSI-certified. See this page for more information about Boost Software License.

## History

#### 3.3.1 (Released on Aug 21, 2018)

- i386 build problem is fixed
- FreeBSD support is added
- Trigonometric functions now evaluate correctly with full FP domain. (PR #210)

#### 3.3 (Released on July 6, 2018)

- AArch64 SVE target support is added (PR #180, #182)
- DFT is now faster (PR #186)
- 3.5-ULP hyperbolic functions are added (PR #192)
- PowerPC VSX target support is added (PR #195)
- Modified Payne-Hanek argument reduction is added to the trigonometric functions in libsleef (PR #197)

#### 3.2 (Released on Feb 26, 2018)

- The whole build system of the project migrated from makefiles to cmake. The makefile build system is now removed.
- GNUABI version of the library with compatibility tests is added.
- Benchmarks that compare `libsleef` vs `SVML` on X86 Linux are available in the project tree under src/libm-benchmarks directory.
- Extensive upstream testing via Travis CI and Appveyor
- log2 is added.
- The library can be compiled to an LLVM bitcode object
- Added masked interface to the library to support AVX512F masked vectorization.
- Use native instructions if available for `sqrt`.
- Removed `libm` dependency.
- fmod(FP remainder), asin, acos, log, pow, log10, exp2, exp10 and log1p functions are now faster.
- Fixed a bug that was making the error of sinpi, cospi, sincospi, and tgamma functions larger than the specifications on very rare occasions.
- Fixed a bug that was preventing the dispatcher from choosing the FMA4 implementation.

#### 3.1 (Released on July 19, 2017)

- Added AArch64 support
- Implemented the remaining C99 math functions : lgamma, tgamma, erf, erfc, fabs, copysign, fmax, fmin, fdim, trunc, floor, ceil, round, rint, modf, ldexp, nextafter, frexp, hypot, and fmod.
- Added dispatcher for x86 functions
- Improved reduction of trigonometric functions
- Added support for 32-bit x86, Cygwin, etc.
- Improved tester
- Etc.

#### 3.0 (Released on Feb. 7, 2017)

- New API is defined
- Functions for DFT are added
- sincospi functions are added
- gencoef now supports single, extended and quad precision in addition to double precision
- Linux, Windows and Mac OS X are supported
- GCC, Clang, Intel Compiler, Microsoft Visual C++ are supported
- The library can be compiled as DLLs
- Files needed for creating a debian package are now included

#### 2.120 (Released on Jan. 30, 2017)

- Relicensed to Boost Software License Version 1.0

#### 2.110 (Released on Dec. 11, 2016)

- The valid range of argument is extended for trig functions
- Specification of each functions regarding to the domain and accuracy is added
- A coefficient generation tool is added
- New testing tools are introduced
- Following functions returned incorrect values when the argument is very large or small : exp, pow, asinh, acosh
- SIMD xsin and xcos returned values more than 1 when FMA is enabled
- Pure C cbrt returned incorrect values when the argument is negative
- tan_u1 returned values with more than 1 ulp of error on rare occasions
- Removed support for Java language(because no one seems using this)

## Related projects and links

- SLEEF is now included in Arm Compiler for HPC.
- Mustafa A. Mohamad is developing a pure Julia port of the SLEEF math library.
- Hal Finkel has adapted SLEEF to PowerPC/QPX. SLEEF is running on Mira IBM BG/Q supercomputer at Argonne National Laboratory.
- SLEEF is adopted in the following projects.
  - MenuetOS
  - RawTherapee
  - John's ECMA-55 Minimal BASIC Compiler
  - Portable Computing Language(pocl)
  - PyTorch
- Intel Short Vector Math Library (SVML) is a commercial library of vectorized math functions, which provides similar functionality to SLEEF. Recently, some of the codes are published as free software, and it is now a part of glibc.