.. _math:

==============
Math Functions
==============

.. include:: ../check.rst

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  log.rst
  stdfix.rst


.. contents:: Table of Contents
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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/>`_.