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.. _math:
==============
Math Functions
==============
.. include:: ../check.rst
.. raw:: html
<style> .green {color:green} </style>
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.. toctree::
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log.rst
stdfix.rst
.. contents:: Table of Contents
:depth: 4
:local:
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>`_.
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.
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
=====================
* To check math functions enabled for Linux:
- `linux-x86_64 <https://github.com/llvm/llvm-project/tree/main/libc/config/linux/x86_64/entrypoints.txt>`_
- `linux-aarch64 <https://github.com/llvm/llvm-project/tree/main/libc/config/linux/aarch64/entrypoints.txt>`_
- `linux-aarch32 <https://github.com/llvm/llvm-project/tree/main/libc/config/linux/arm/entrypoints.txt>`_
- `linux-riscv64 <https://github.com/llvm/llvm-project/tree/main/libc/config/linux/riscv64/entrypoints.txt>`_
* To check math functions enabled for Windows:
- `windows-x86_64 <https://github.com/llvm/llvm-project/tree/main/libc/config/windows/entrypoints.txt>`_
- windows-aarch64 - to be added
* To check math functions enabled for macOS:
- `darwin-x86_64 <https://github.com/llvm/llvm-project/tree/main/libc/config/darwin/x86_64/entrypoints.txt>`_
- `darwin-aarch64 <https://github.com/llvm/llvm-project/tree/main/libc/config/darwin/arm/entrypoints.txt>`_
* To check math functions enabled for GPU:
- `gpu-entrypoints <https://github.com/llvm/llvm-project/tree/main/libc/config/gpu/entrypoints.txt>`_
* To check math functions enabled for embedded system:
- `baremetal-aarch32 <https://github.com/llvm/llvm-project/tree/main/libc/config/baremetal/arm/entrypoints.txt>`_
- baremetal-riscv32 - to be added
Basic Operations
================
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| <Func> | <Func_f> (float) | <Func> (double) | <Func_l> (long double) | <Func_f16> (float16) | <Func_f128> (float128) | C23 Definition Section | C23 Error Handling Section |
+==================+==================+=================+========================+======================+========================+========================+============================+
| ceil | |check| | |check| | |check| | | |check| | 7.12.9.1 | F.10.6.1 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| canonicalize | |check| | |check| | |check| | | |check| | 7.12.11.7 | F.10.8.7 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| copysign | |check| | |check| | |check| | | |check| | 7.12.11.1 | F.10.8.1 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| dadd | N/A | N/A | | N/A | | 7.12.14.1 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| ddiv | N/A | N/A | | N/A | | 7.12.14.4 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| dfma | N/A | N/A | | N/A | | 7.12.14.5 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| dmul | N/A | N/A | | N/A | | 7.12.14.3 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| dsub | N/A | N/A | | N/A | | 7.12.14.2 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fabs | |check| | |check| | |check| | | |check| | 7.12.7.3 | F.10.4.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fadd | N/A | | | N/A | | 7.12.14.1 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fdim | |check| | |check| | |check| | | |check| | 7.12.12.1 | F.10.9.1 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fdiv | N/A | | | N/A | | 7.12.14.4 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| ffma | N/A | | | N/A | | 7.12.14.5 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| floor | |check| | |check| | |check| | | |check| | 7.12.9.2 | F.10.6.2 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fmax | |check| | |check| | |check| | | |check| | 7.12.12.2 | F.10.9.2 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fmaximum | |check| | |check| | |check| | | |check| | 7.12.12.4 | F.10.9.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fmaximum_mag | |check| | |check| | |check| | | |check| | 7.12.12.6 | F.10.9.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fmaximum_mag_num | |check| | |check| | |check| | | |check| | 7.12.12.10 | F.10.9.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fmaximum_num | |check| | |check| | |check| | | |check| | 7.12.12.8 | F.10.9.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fmin | |check| | |check| | |check| | | |check| | 7.12.12.3 | F.10.9.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fminimum | |check| | |check| | |check| | | |check| | 7.12.12.5 | F.10.9.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fminimum_mag | |check| | |check| | |check| | | |check| | 7.12.12.7 | F.10.9.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fminimum_mag_num | |check| | |check| | |check| | | |check| | 7.12.12.11 | F.10.9.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fminimum_num | |check| | |check| | |check| | | |check| | 7.12.12.9 | F.10.9.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fmod | |check| | |check| | |check| | | |check| | 7.12.10.1 | F.10.7.1 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fmul | N/A | | | N/A | | 7.12.14.3 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| frexp | |check| | |check| | |check| | | |check| | 7.12.6.7 | F.10.3.7 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fromfp | |check| | |check| | |check| | | |check| | 7.12.9.10 | F.10.6.10 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fromfpx | |check| | |check| | |check| | | |check| | 7.12.9.11 | F.10.6.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fsub | N/A | | | N/A | | 7.12.14.2 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| ilogb | |check| | |check| | |check| | | |check| | 7.12.6.8 | F.10.3.8 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| ldexp | |check| | |check| | |check| | | |check| | 7.12.6.9 | F.10.3.9 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| llogb | |check| | |check| | |check| | | |check| | 7.12.6.10 | F.10.3.10 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| llrint | |check| | |check| | |check| | | |check| | 7.12.9.5 | F.10.6.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| llround | |check| | |check| | |check| | | |check| | 7.12.9.7 | F.10.6.7 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| logb | |check| | |check| | |check| | | |check| | 7.12.6.17 | F.10.3.17 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| lrint | |check| | |check| | |check| | | |check| | 7.12.9.5 | F.10.6.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| lround | |check| | |check| | |check| | | |check| | 7.12.9.7 | F.10.6.7 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| modf | |check| | |check| | |check| | | |check| | 7.12.6.18 | F.10.3.18 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| nan | |check| | |check| | |check| | | |check| | 7.12.11.2 | F.10.8.2 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| nearbyint | |check| | |check| | |check| | | | 7.12.9.3 | F.10.6.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| nextafter | |check| | |check| | |check| | | |check| | 7.12.11.3 | F.10.8.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| nextdown | |check| | |check| | |check| | | |check| | 7.12.11.6 | F.10.8.6 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| nexttoward | |check| | |check| | |check| | | N/A | 7.12.11.4 | F.10.8.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| nextup | |check| | |check| | |check| | | |check| | 7.12.11.5 | F.10.8.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| remainder | |check| | |check| | |check| | | | 7.12.10.2 | F.10.7.2 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| remquo | |check| | |check| | |check| | | | 7.12.10.3 | F.10.7.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| rint | |check| | |check| | |check| | | |check| | 7.12.9.4 | F.10.6.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| round | |check| | |check| | |check| | | |check| | 7.12.9.6 | F.10.6.6 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| roundeven | |check| | |check| | |check| | | |check| | 7.12.9.8 | F.10.6.8 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| scalbn | |check| | |check| | |check| | | | 7.12.6.19 | F.10.3.19 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| trunc | |check| | |check| | |check| | | |check| | 7.12.9.9 | F.10.6.9 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| ufromfp | |check| | |check| | |check| | | |check| | 7.12.9.10 | F.10.6.10 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| ufromfpx | |check| | |check| | |check| | | |check| | 7.12.9.11 | F.10.6.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
Higher Math Functions
=====================
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| <Func> | <Func_f> (float) | <Func> (double) | <Func_l> (long double) | <Func_f16> (float16) | <Func_f128> (float128) | C23 Definition Section | C23 Error Handling Section |
+===========+==================+=================+========================+======================+========================+========================+============================+
| acos | |check| | | | | | 7.12.4.1 | F.10.1.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| acosh | |check| | | | | | 7.12.5.1 | F.10.2.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| acospi | | | | | | 7.12.4.8 | F.10.1.8 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| asin | |check| | | | | | 7.12.4.2 | F.10.1.2 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| asinh | |check| | | | | | 7.12.5.2 | F.10.2.2 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| asinpi | | | | | | 7.12.4.9 | F.10.1.9 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| atan | |check| | | | | | 7.12.4.3 | F.10.1.3 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| atan2 | |check| | | | | | 7.12.4.4 | F.10.1.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| atan2pi | | | | | | 7.12.4.11 | F.10.1.11 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| atanh | |check| | | | | | 7.12.5.3 | F.10.2.3 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| atanpi | | | | | | 7.12.4.10 | F.10.1.10 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| cbrt | | | | | | 7.12.7.1 | F.10.4.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| compoundn | | | | | | 7.12.7.2 | F.10.4.2 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| cos | |check| | large | | | | 7.12.4.5 | F.10.1.5 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| cosh | |check| | | | | | 7.12.5.4 | F.10.2.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| cospi | | | | | | 7.12.4.12 | F.10.1.12 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| dsqrt | N/A | N/A | | N/A | | 7.12.14.6 | F.10.11 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| erf | |check| | | | | | 7.12.8.1 | F.10.5.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| erfc | | | | | | 7.12.8.2 | F.10.5.2 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| exp | |check| | |check| | | | | 7.12.6.1 | F.10.3.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| exp10 | |check| | |check| | | | | 7.12.6.2 | F.10.3.2 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| exp10m1 | | | | | | 7.12.6.3 | F.10.3.3 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| exp2 | |check| | |check| | | | | 7.12.6.4 | F.10.3.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| exp2m1 | |check| | | | | | 7.12.6.5 | F.10.3.5 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| expm1 | |check| | |check| | | | | 7.12.6.6 | F.10.3.6 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fma | |check| | |check| | | | | 7.12.13.1 | F.10.10.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| fsqrt | N/A | | | N/A | | 7.12.14.6 | F.10.11 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| hypot | |check| | |check| | | | | 7.12.7.4 | F.10.4.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| lgamma | | | | | | 7.12.8.3 | F.10.5.3 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| log | |check| | |check| | | | | 7.12.6.11 | F.10.3.11 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| log10 | |check| | |check| | | | | 7.12.6.12 | F.10.3.12 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| log10p1 | | | | | | 7.12.6.13 | F.10.3.13 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| log1p | |check| | |check| | | | | 7.12.6.14 | F.10.3.14 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| log2 | |check| | |check| | | | | 7.12.6.15 | F.10.3.15 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| log2p1 | | | | | | 7.12.6.16 | F.10.3.16 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| logp1 | | | | | | 7.12.6.14 | F.10.3.14 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| pow | |check| | | | | | 7.12.7.5 | F.10.4.5 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| pown | | | | | | 7.12.7.6 | F.10.4.6 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| powr | | | | | | 7.12.7.7 | F.10.4.7 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| rootn | | | | | | 7.12.7.8 | F.10.4.8 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| rsqrt | | | | | | 7.12.7.9 | F.10.4.9 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| sin | |check| | large | | | | 7.12.4.6 | F.10.1.6 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| sincos | |check| | large | | | | | |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| sinh | |check| | | | | | 7.12.5.5 | F.10.2.5 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| sinpi | | | | | | 7.12.4.13 | F.10.1.13 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| sqrt | |check| | |check| | |check| | | |check| | 7.12.7.10 | F.10.4.10 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| tan | |check| | | | | | 7.12.4.7 | F.10.1.7 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| tanh | |check| | | | | | 7.12.5.6 | F.10.2.6 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| tanpi | | | | | | 7.12.4.14 | F.10.1.14 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
| tgamma | | | | | | 7.12.8.4 | F.10.5.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
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.
* N/A: Not defined in the standard or will not be added.
..
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/performance_testing <https://github.com/llvm/llvm-project/tree/main/libc/test/src/math/performance_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). Fmod performance results obtained with "performance_testing".
+--------------+-------------------------------+-------------------------------+-------------------------------------+----------------------------------------------------------------------+
| <Func> | Reciprocal throughput (clk) | Latency (clk) | Testing ranges | Testing configuration |
| +-----------+-------------------+-----------+-------------------+ +-------------+-------------------------+--------------+---------------+
| | LLVM libc | Reference (glibc) | LLVM libc | Reference (glibc) | | CPU | OS | Compiler | Special flags |
+==============+===========+===================+===========+===================+=====================================+=============+=========================+==============+===============+
| acosf | 24 | 29 | 62 | 77 | :math:`[-1, 1]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| acoshf | 18 | 26 | 73 | 74 | :math:`[1, 21]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| asinf | 23 | 27 | 62 | 62 | :math:`[-1, 1]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| asinhf | 21 | 39 | 77 | 91 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| atanf | 27 | 29 | 79 | 68 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| atanhf | 18 | 66 | 68 | 133 | :math:`[-1, 1]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| cosf | 13 | 32 | 53 | 59 | :math:`[0, 2\pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| coshf | 14 | 20 | 50 | 48 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| expf | 9 | 7 | 44 | 38 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| exp10f | 10 | 8 | 40 | 38 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| exp2f | 9 | 6 | 35 | 31 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| expm1f | 9 | 44 | 42 | 121 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| fmodf | 73 | 263 | - | - | [MIN_NORMAL, MAX_NORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
| +-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| | 9 | 11 | - | - | [0, MAX_SUBNORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| fmod | 595 | 3297 | - | - | [MIN_NORMAL, MAX_NORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
| +-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| | 14 | 13 | - | - | [0, MAX_SUBNORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| 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 | 9 | 17 | 35 | 48 | :math:`[e^{-1}, e]` | Ryzen 5900X | Ubuntu 22.04 LTS x86_64 | Clang 15.0.6 | 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 | 12 | 25 | 51 | 57 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| sincosf | 19 | 30 | 57 | 68 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| sinhf | 13 | 63 | 48 | 137 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| tanf | 16 | 50 | 61 | 107 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
| tanhf | 13 | 55 | 57 | 123 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 22.04 LTS x86_64 | Clang 14.0.0 | FMA |
+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+-------------+-------------------------+--------------+---------------+
Algorithms + Implementation Details
===================================
* :doc:`log`
Fixed-point Arithmetics
=======================
* :doc:`stdfix`
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/>`_.