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llvm-project/libc/docs/math.rst
Kirill Okhotnikov b8e8012aa2 [libc][math] fmod/fmodf implementation. 
This is a implementation of find remainder fmod function from standard libm.
The underline algorithm is developed by myself, but probably it was first
invented before.
Some features of the implementation:
1. The code is written on more-or-less modern C++.
2. One general implementation for both float and double precision numbers.
3. Spitted platform/architecture dependent and independent code and tests.
4. Tests covers 100% of the code for both float and double numbers. Tests cases with NaN/Inf etc is copied from glibc.
5. The new implementation in general 2-4 times faster for “regular” x,y values. It can be 20 times faster for x/y huge value, but can also be 2 times slower for double denormalized range (according to perf tests provided).
6. Two different implementation of division loop are provided. In some platforms division can be very time consuming operation. Depend on platform it can be 3-10 times slower than multiplication.

Performance tests:

The test is based on core-math project (https://gitlab.inria.fr/core-math/core-math). By Tue Ly suggestion I took hypot function and use it as template for fmod. Preserving all test cases.

`./check.sh <--special|--worst> fmodf` passed.
`CORE_MATH_PERF_MODE=rdtsc ./perf.sh fmodf` results are

```
GNU libc version: 2.35
GNU libc release: stable
21.166 <-- FPU
51.031 <-- current glibc
37.659 <-- this fmod version.
```
2022-06-24 23:09:14 +02:00

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===========================
Math Functions in LLVM-libc
===========================
.. role:: raw-html(raw)
:format: html
.. |check| replace:: :raw-html:`&#x2705`
.. contents:: Table of Contents
:depth: 4
:local:
Summary
=======
* This document tracks the status of the implementation of math functions in
LLVM libc.
Implementation Requirements / Goals
===================================
* The highest priority is to be as accurate as possible, according to the C and
IEEE 754 standards. By default, we will aim to be correctly rounded for `all rounding modes <https://en.cppreference.com/w/c/numeric/fenv/FE_round>`_.
The current rounding mode of the floating point environment is used to perform
computations and produce the final results.
- To test for correctness, we compare the outputs with other correctly rounded
multiple-precision math libraries such as the `GNU MPFR library <https://www.mpfr.org/>`_
or the `CORE-MATH library <https://core-math.gitlabpages.inria.fr/>`_.
* Our next requirement is that the outputs are consistent across all platforms.
Notice that the consistency requirement will be satisfied automatically if the
implementation is correctly rounded.
* Our last requirement for the implementations is to have good and predicable
performance:
- The average performance should be comparable to other ``libc``
implementations.
- The worst case performance should be within 10X-20X of the average.
- Platform-specific implementations or instructions could be added whenever it
makes sense and provides significant performance boost.
* For other use cases that have strict requirements on the code size, memory
footprint, or latency, such as embedded systems, we will aim to be as accurate
as possible within the memory or latency budgets, and consistent across all
platforms.
Source Locations
================
- The main source is located at: `libc/src/math <https://github.com/llvm/llvm-project/tree/main/libc/src/math>`_.
- The tests are located at: `libc/test/src/math <https://github.com/llvm/llvm-project/tree/main/libc/test/src/math>`_.
- The floating point utilities are located at: `libc/src/__support/FPUtil <https://github.com/llvm/llvm-project/tree/main/libc/src/__support/FPUtil>`_.
Add a new math function to LLVM libc
====================================
* To add a new math function, follow the steps at: `libc/src/math/docs/add_math_function.md <https://github.com/llvm/llvm-project/tree/main/libc/src/math/docs/add_math_function.md>`_.
Implementation Status
=====================
Basic Operations
----------------
============== ================ =============== ======================
<Func> <Func_f> (float) <Func> (double) <Func_l> (long double)
============== ================ =============== ======================
ceil |check| |check| |check|
copysign |check| |check| |check|
fabs |check| |check| |check|
fdim |check| |check| |check|
floor |check| |check| |check|
fmax |check| |check| |check|
fmin |check| |check| |check|
fmod |check| |check|
fpclassify
frexp |check| |check| |check|
ilogb |check| |check| |check|
isfinite
isgreater
isgreaterequal
isinf
isless
islessequal
islessgreater
isnan
isnormal
isubordered
ldexp |check| |check| |check|
llrint |check| |check| |check|
llround |check| |check| |check|
logb |check| |check| |check|
lrint |check| |check| |check|
lround |check| |check| |check|
modf |check| |check| |check|
nan
nearbyint |check| |check| |check|
nextafter |check| |check| |check|
nexttoward
remainder |check| |check| |check|
remquo |check| |check| |check|
rint |check| |check| |check|
round |check| |check| |check|
scalbn
signbit
trunc |check| |check| |check|
============== ================ =============== ======================
Higher Math Functions
---------------------
============== ================ =============== ======================
<Func> <Func_f> (float) <Func> (double) <Func_l> (long double)
============== ================ =============== ======================
acos
acosh
asin
asinh
atan
atan2
atanh
cbrt
cos |check| |check|
cosh
erf
erfc
exp |check|
exp2 |check|
expm1 |check|
fma |check| |check|
hypot |check| |check|
lgamma
log |check|
log10 |check|
log1p |check|
log2 |check|
pow
sin |check| |check|
sincos |check| |check|
sinh
sqrt |check| |check| |check|
tan
tanh
tgamma
============== ================ =============== ======================
Accuracy of Higher Math Functions
=================================
============== ================ =============== ======================
<Func> <Func_f> (float) <Func> (double) <Func_l> (long double)
============== ================ =============== ======================
cos 0.776 ULPs large
exp |check|
exp2 |check|
expm1 |check|
fma |check| |check|
hypot |check| |check|
log |check|
log10 |check|
log1p |check|
log2 |check|
sin 0.561 ULPs large
sincos 0.776 ULPs large
sqrt |check| |check| |check|
============== ================ =============== ======================
Legends:
* |check|: correctly rounded for all 4 rounding modes.
* CR: correctly rounded for the default rounding mode (round-to-the-nearest,
tie-to-even).
* x ULPs: largest errors recorded.
..
TODO(lntue): Add a new page to discuss about the algorithms used in the
implementations and include the link here.
Performance
===========
* Simple performance testings are located at: `libc/test/src/math/differential_testing <https://github.com/llvm/llvm-project/tree/main/libc/test/src/math/differential_testing>`_.
* We also use the *perf* tool from the `CORE-MATH <https://core-math.gitlabpages.inria.fr/>`_
project: `link <https://gitlab.inria.fr/core-math/core-math/-/tree/master>`_.
The performance results from the CORE-MATH's perf tool are reported in the
table below, using the system library as reference (such as the `GNU C library <https://www.gnu.org/software/libc/>`_
on Linux).
+--------------+-------------------------------+-------------------------------+-------------------------------------+---------------------------------------------------------------------+
| <Func> | Reciprocal throughput (ns) | Latency (ns) | Testing ranges | Testing configuration |
| +-----------+-------------------+-----------+-------------------+ +------------+-------------------------+--------------+---------------+
| | LLVM libc | Reference (glibc) | LLVM libc | Reference (glibc) | | CPU | OS | Compiler | Special flags |
+==============+===========+===================+===========+===================+=====================================+============+=========================+==============+===============+
| cosf | 37 | 32 | 73 | 72 | :math:`[0, 2\pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| expf | 14 | 9 | 58 | 42 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| exp2f | 25 | 8 | 81 | 37 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| expm1f | 14 | 53 | 59 | 146 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| hypotf | 25 | 15 | 64 | 49 | :math:`[-10, 10] \times [-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| logf | 12 | 10 | 56 | 46 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| log10f | 13 | 25 | 57 | 72 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| log1pf | 16 | 33 | 61 | 97 | :math:`[e^{-0.5} - 1, e^{0.5} - 1]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| log2f | 13 | 10 | 57 | 46 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
| sinf | 36 | 31 | 72 | 71 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
References
==========
* `CRLIBM <https://hal-ens-lyon.archives-ouvertes.fr/ensl-01529804/file/crlibm.pdf>`_.
* `RLIBM <https://people.cs.rutgers.edu/~sn349/rlibm/>`_.
* `Sollya <https://www.sollya.org/>`_.
* `The CORE-MATH Project <https://core-math.gitlabpages.inria.fr/>`_.
* `The GNU C Library (glibc) <https://www.gnu.org/software/libc/>`_.
* `The GNU MPFR Library <https://www.mpfr.org/>`_.
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