[libc][NFC] Move aligned access implementations to separate header

[llvm-project.git] / libc / docs / math / index.rst

.. _math:

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

.. include:: ../check.rst

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  log.rst


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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:

  - `barebone-aarch32 <https://github.com/llvm/llvm-project/tree/main/libc/config/baremetal/arm/entrypoints.txt>`_ 

  - barebone-riscv32 - to be added

Basic Operations
----------------

+--------------+---------------------------------------+-------------------+-------------------+-------------------+-------------------+
| <Func>       |  Linux                                | Windows           | MacOS             | Embedded          | GPU               |
|              +---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
|              | x86_64  | aarch64 | aarch32 | riscv64 | x86_64  | aarch64 | x86_64  | aarch64 | aarch32 | riscv32 | AMD     | nVidia  |
+==============+=========+=========+=========+=========+=========+=========+=========+=========+=========+=========+=========+=========+
| ceil         | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ceilf        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ceill        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| copysign     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| copysignf    | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| copysignl    | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fabs         | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fabsf        | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fabsl        | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fdim         | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fdimf        | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fdiml        | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| floor        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| floorf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| floorl       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmax         | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmaxf        | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmaxl        | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmin         | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fminf        | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fminl        | |check| | |check| | |check| | |check| | |check| |         |         | |check| | |check| |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmod         | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmodf        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmodl        |         |         |         |         |         |         |         |         |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| frexp        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| frexpf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| frexpl       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ilogb        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ilogbf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ilogbl       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ldexp        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ldexpf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| ldexpl       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| llrint       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| llrintf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| llrintl      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| llround      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| llroundf     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| llroundl     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| logb         | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| logbf        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| logbl        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lrint        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lrintf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lrintl       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lround       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lroundf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lroundl      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| modf         | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| modff        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| modfl        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nan          |         |         |         |         |         |         |         |         |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nanf         |         |         |         |         |         |         |         |         |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nanl         |         |         |         |         |         |         |         |         |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nearbyint    | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nearbyintf   | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nearbyintl   | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nextafter    | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nextafterf   | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nextafterl   | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nexttoward   |         |         |         |         |         |         |         |         |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nexttowardf  |         |         |         |         |         |         |         |         |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| nexttowardl  |         |         |         |         |         |         |         |         |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remainder    | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remainderf   | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remainderl   | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remquo       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remquof      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| remquol      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| rint         | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| rintf        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| rintl        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| round        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| roundf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| roundl       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| scalbn       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| scalbnf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| scalbnl      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| trunc        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| truncf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| truncl       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+--------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+


Higher Math Functions
---------------------

+------------+---------------------------------------+-------------------+-------------------+-------------------+-------------------+
| <Func>     |  Linux                                | Windows           | MacOS             | Embedded          | GPU               |
|            +---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
|            | x86_64  | aarch64 | aarch32 | riscv64 | x86_64  | aarch64 | x86_64  | aarch64 | aarch32 | riscv32 | AMD     | nVidia  |
+============+=========+=========+=========+=========+=========+=========+=========+=========+=========+=========+=========+=========+
| acos       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acosf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acosl      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acosh      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acoshf     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| acoshl     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asin       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinl      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinh      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinhf     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| asinhl     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atan       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanl      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atan2      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atan2f     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atan2l     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanh      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanhf     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| atanhl     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cbrt       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cbrtf      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cbrtl      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cos        | |check| |         |         |         | |check| |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cosf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cosl       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| cosh       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| coshf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| coshl      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erf        |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erff       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfl       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfc       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfcf      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfcl      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp        |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expl       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp10      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp10f     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp10l     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp2       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp2f      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| exp2l      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expm1      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expm1f     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| expm1l     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fma        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmaf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| fmal       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| hypot      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| hypotf     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| hypotl     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lgamma     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lgammaf    |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| lgammal    |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log        | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| logf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| logl       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log10      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log10f     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log10l     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log1p      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log1pf     | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log1pl     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log2       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log2f      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| log2l      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| pow        |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| powf       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| powl       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sin        | |check| |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinl       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sincos     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sincosf    | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sincosl    |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinh       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinhf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sinhl      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sqrt       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sqrtf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| sqrtl      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tan        | |check| |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanf       | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanl       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanh       |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanhf      | |check| | |check| |         | |check| | |check| |         |         | |check| |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tanhl      |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tgamma     |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tgammaf    |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| tgammal    |         |         |         |         |         |         |         |         |         |         |         |         |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+


Accuracy of Higher Math Functions
=================================

============== ================ =============== ======================
<Func>         <Func_f> (float) <Func> (double) <Func_l> (long double)
============== ================ =============== ======================
acos           |check|
acosh          |check|
asin           |check|
asinh          |check|
atan           |check|
atanh          |check|
cos            |check|          large
cosh           |check|
erf            |check|
exp            |check|
exp10          |check|
exp2           |check|
expm1          |check|
fma            |check|          |check|
hypot          |check|          |check|
log            |check|          |check|
log10          |check|          |check|
log1p          |check|          |check|
log2           |check|          |check|
sin            |check|          large
sincos         |check|          large
sinh           |check|
sqrt           |check|          |check|         |check|
tan            |check|
tanh           |check|
============== ================ =============== ======================


Legends:

* |check| : correctly rounded for all 4 rounding modes.
* CR: correctly rounded for the default rounding mode (round-to-the-nearest,
  tie-to-even).
* x ULPs: largest errors recorded.

..
  TODO(lntue): Add a new page to discuss about the algorithms used in the
  implementations and include the link here.


Performance
===========

* Simple performance testings are located at: `libc/test/src/math/differential_testing <https://github.com/llvm/llvm-project/tree/main/libc/test/src/math/differential_testing>`_.

* We also use the *perf* tool from the `CORE-MATH <https://core-math.gitlabpages.inria.fr/>`_
  project: `link <https://gitlab.inria.fr/core-math/core-math/-/tree/master>`_.
  The performance results from the CORE-MATH's perf tool are reported in the
  table below, using the system library as reference (such as the `GNU C library <https://www.gnu.org/software/libc/>`_
  on Linux). Fmod performance results obtained with "differential_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`

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/>`_.