Highway-Sleef

Sleef is one of the best vectorized math libraries, created by Professor Naoki Shibata (Nara Institue of Science and Technology). In my opinion, it remains under-utilized due to difficulties in integrating it in downstream projects.

On the other hand, Highway is a best-in-class portable SIMD library in C++, offering some of the easiest runtime feature detection and dispatching of any SIMD library.

This project aims to expand and improve the math functionality of Highway by performing semi-automated translation of the excellent math implementations from Sleef.

Building

Initialize git submodules:

git submodule update --init

To get LTO for sleef to work, I've configured with cmake as follows:

CC=clang CXX=clang++ cmake -B build -S . -G Ninja -DLLVM_AR_COMMAND=llvm-ar-14

Replace llvm-ar-14 with the appropriate version number present on your machine

To run performance or correctness tests:

cmake --build build
./build/src/tests/identical_to_sleef
./build/src/tests/measure_ulp
./build/src/tests/measure_speed

Example results of these tests are included in outputs, along with a copy of the generated result outputs/sleef-generated.h

Results

Current status:

• All of highway's single-precision functions with 1 argument have translated equivalents from sleef
• All results from the translation are bitwise identical to original SLEEF, with the exception of some NaN outputs (I've noticed some sign-mismatches on NaN's which seem to be dependent on optimization level)
  • This is only with testing on AVX2, so is not guaranteed to hold in other instruction sets
• See src/gen-bindings/README.md for a description of the translation code

Next steps:

• Implement the rest of the SLEEF operations including double-precision
  • See the SLEEF documentation for a full list of these operations. Highlights include trig functions with arguments automatically multiplied by pi, along with error and gamma function
• Develop testing methodology for functions that cannot be exhaustively tested for all inputs
• Test accuracy and performance on more platforms and Hwy configurations
• Try to merge into upstream hwy library.

Accuracy

Notes:

• hwy sometimes produces a NaN when the correct result is not NaN. These inputs have been omitted for comparison.
• Inputs that cause an Inf when the correct result is not Inf cause Inf to be marked as ULP.
• For many trig functions, SLEEF provides a high and low precision variant
• In a few cases, SLEEF has a more restrictive valid range than hwy, which are marked accordingly

Precision in valid range

Op	valid range	Hwy	Sleef 1 ULP	Sleef 3.5 ULP
Exp	[-FLT_MAX, 104]	1.03	0.988	N/A
Expm1	[-FLT_MAX, 104]	3.55	1	N/A
Log	(0, FLT_MAX]	0.995	0.995	N/A
Log1p	(0, FLT_MAX] for hwy (0, 1e38] for sleef	2.47	1	N/A
Log2	(0, FLT_MAX]	1.87	0.992	N/A
Sin	[-39000, +39000]	2.47	0.904	2.47
Cos	[-39000, +39000]	2.42	0.936	2.49
Tan	[-39000, 39000]	N/A	1.4	3.46
Sinh	[-88.722801,88.722801] [-88, 88] for Sleef 3.5 ULP	3.4	1	3.4
Cosh	[-88, 88]	N/A	0.503	1.45
Tanh	[-FLT_MAX, FLT_MAX]	2.86	1	2.86
Asin	[-1, 1]	2.47	0.981	2.43
Acos	[-1, 1]	1.28	0.694	1.28
Atan	[-FLT_MAX, FLT_MAX]	2.65	0.938	2.64
Asinh	[-FLT_MAX, FLT_MAX] for hwy [sqrt(-FLT_MAX), sqrt(FLT_MAX)] for sleef	3.27	1	N/A
Acosh	[1, FLT_MAX] for hwy [1, sqrt(FLT_MAX)] for sleef	3.43	0.512	N/A
Atanh	[-1, 1]	3.22	1	N/A

Precision (all floats)

Op	Hwy	Sleef 1 ULP	Sleef 3.5 ULP
Exp	1.03	0.988	N/A
Expm1	3.55	1	N/A
Log	0.995	0.995	N/A
Log1p	2.47	Inf	N/A
Log2	1.87	0.992	N/A
Sin	Inf	1.09	2.47
Cos	Inf	1.07	2.49
Tan	N/A	1.4	3.46
Sinh	3.4	Inf	Inf
Cosh	N/A	Inf	Inf
Tanh	2.86	1	2.86
Asin	2.47	0.981	2.43
Acos	1.28	0.694	1.28
Atan	2.65	0.938	2.64
Asinh	3.27	Inf	N/A
Acosh	3.43	Inf	N/A
Atanh	3.22	1	N/A

Speed

Notes:

• Sleef runs more slowly on large inputs methods for sin, cos, and tan. The 1 ULP has a single fallback method, while 3.5 ULP has two fallback methods. The benchmarks for these functions show both a narrow range (-125, 125) and a wide range [-39000, 39000] in the timing table
• The translated version of Sleef sometimes runs more slowly than the original Sleef. One cause I have found for this is that NearestInt in hwy performs additional checks sleef can omit without any loss in mathematical precision.
• All results were measured using AVX2 on a Framework laptop with i5-1240P processor. The benchmark task was to repeatedly apply the function on 4,096 random floats with memory clobbering between iterations to avoid optimizing out the code

Units are ns/float, then time relative to best.

Op	Range tested	Hwy	Translated 1 ULP	Sleef 1 ULP	Translated 3.5 ULP	Sleef 3.5 ULP
Exp	[-FLT_MAX, 104]	0.5 ns (127%)	0.4 ns (118%)	0.4 ns (100%)	N/A	N/A
Expm1	[-FLT_MAX, 104]	0.6 ns (100%)	2.1 ns (375%)	1.8 ns (322%)	N/A	N/A
Log	(0, FLT_MAX]	0.4 ns (100%)	1.2 ns (279%)	1.2 ns (264%)	N/A	N/A
Log1p	[0, 1e38]	0.4 ns (100%)	1.0 ns (240%)	1.0 ns (223%)	N/A	N/A
Log2	(0, FLT_MAX]	0.4 ns (100%)	1.2 ns (286%)	1.2 ns (280%)	N/A	N/A
Sin	(-125, 125)	0.5 ns (118%)	0.9 ns (239%)	0.8 ns (214%)	0.4 ns (100%)	0.4 ns (103%)
Sin	[-39000, 39000]	0.3 ns (100%)	3.7 ns (1080%)	3.5 ns (1018%)	0.6 ns (161%)	0.7 ns (205%)
Cos	(-125, 125)	0.5 ns (121%)	1.1 ns (266%)	0.9 ns (222%)	0.4 ns (100%)	0.4 ns (102%)
Cos v2	[-39000, 39000]	0.3 ns (100%)	4.2 ns (1228%)	4.6 ns (1349%)	0.6 ns (191%)	0.6 ns (181%)
Tan	(-125, 125)	N/A	4.2 ns (619%)	4.5 ns (669%)	0.7 ns (100%)	0.7 ns (107%)
Tan v2	[-39000, 39000]	N/A	1.6 ns (239%)	1.6 ns (234%)	0.7 ns (100%)	0.8 ns (118%)
Sinh	[-88, 88]	0.8 ns (100%)	8.0 ns (1056%)	8.1 ns (1061%)	0.9 ns (119%)	0.8 ns (106%)
Cosh	[-88, 88]	N/A	8.7 ns (387%)	7.4 ns (332%)	2.2 ns (100%)	3.2 ns (141%)
Tanh	[-FLT_MAX, FLT_MAX]	0.9 ns (124%)	2.6 ns (362%)	2.0 ns (281%)	0.9 ns (119%)	0.7 ns (100%)
Asin	[-1, 1]	0.4 ns (106%)	1.1 ns (309%)	1.1 ns (309%)	0.4 ns (100%)	0.4 ns (102%)
Acos	[-1, 1]	0.5 ns (100%)	1.5 ns (282%)	1.4 ns (268%)	0.6 ns (113%)	0.6 ns (118%)
Atan	[-FLT_MAX, FLT_MAX]	5.6 ns (100%)	33.2 ns (594%)	36.1 ns (646%)	6.3 ns (114%)	6.0 ns (107%)
Asinh	[-sqrt(FLT_MAX), sqrt(FLT_MAX)]	1.4 ns (100%)	44.7 ns (3287%)	41.9 ns (3076%)	N/A	N/A
Acosh	[1, sqrt(FLT_MAX)]	1.2 ns (100%)	3.1 ns (258%)	3.2 ns (261%)	N/A	N/A
Atanh	(-1, 1)	0.8 ns (100%)	2.1 ns (269%)	2.3 ns (289%)	N/A	N/A