.. _math:
======
math.h
======
.. include:: ../../check.rst
.. raw:: html
.. role:: green
.. toctree::
:hidden:
log.rst
.. contents:: Table of Contents
:depth: 4
:local:
Source Locations
================
- The main source is located at: `libc/src/math `_.
- The tests are located at: `libc/test/src/math `_.
- The floating point utilities are located at: `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 `_.
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 `_
or the `CORE-MATH library `_.
* 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 `_.
Implementation Status
=====================
* To check math functions enabled for Linux:
- `linux-x86_64 `_
- `linux-aarch64 `_
- `linux-aarch32 `_
- `linux-riscv and linux-riscv32 `_
* To check math functions enabled for Windows:
- `windows-x86_64 `_
- windows-aarch64 - to be added
* To check math functions enabled for macOS:
- `darwin-x86_64 `_
- `darwin-aarch64 `_
* To check math functions enabled for GPU:
- `gpu-entrypoints `_
* To check math functions enabled for embedded system:
- `baremetal-aarch32 `_
- baremetal-riscv32 - to be added
Basic Operations
================
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| | (float) | (double) | (long double) | (float16) | (float128) | (bfloat16) | C23 Definition Section | C23 Error Handling Section |
+==================+==================+=================+========================+======================+========================+========================+========================+============================+
| ceil | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.1 | F.10.6.1 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| canonicalize | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.11.7 | F.10.8.7 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| copysign | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.11.1 | F.10.8.1 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| dadd | N/A | N/A | |check| | N/A | |check|\* | N/A | 7.12.14.1 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| ddiv | N/A | N/A | |check| | N/A | |check|\* | N/A | 7.12.14.4 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| dfma | N/A | N/A | |check| | N/A | |check|\* | N/A | 7.12.14.5 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| dmul | N/A | N/A | |check| | N/A | |check|\* | N/A | 7.12.14.3 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| dsub | N/A | N/A | |check| | N/A | |check|\* | N/A | 7.12.14.2 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| f16add | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.1 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| f16div | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.4 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| f16fma | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.5 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| f16mul | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.3 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| f16sub | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.2 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| bf16add | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.1 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| bf16div | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.4 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| bf16fma | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.5 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| bf16mul | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.3 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| bf16sub | |check|\* | |check|\* | |check|\* | N/A | |check| | N/A | 7.12.14.2 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fabs | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.7.3 | F.10.4.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fadd | N/A | |check| | |check| | N/A | |check| | N/A | 7.12.14.1 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fdim | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.1 | F.10.9.1 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fdiv | N/A | |check| | |check| | N/A | |check|\* | N/A | 7.12.14.4 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| ffma | N/A | |check| | |check| | N/A | |check|\* | N/A | 7.12.14.5 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| floor | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.2 | F.10.6.2 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fmax | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.2 | F.10.9.2 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fmaximum | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.4 | F.10.9.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fmaximum_mag | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.6 | F.10.9.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fmaximum_mag_num | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.10 | F.10.9.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fmaximum_num | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.8 | F.10.9.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fmin | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.3 | F.10.9.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fminimum | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.5 | F.10.9.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fminimum_mag | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.7 | F.10.9.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fminimum_mag_num | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.11 | F.10.9.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fminimum_num | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.12.9 | F.10.9.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fmod | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.10.1 | F.10.7.1 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fmul | N/A | |check| | |check| | N/A | |check|\* | N/A | 7.12.14.3 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| frexp | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.6.7 | F.10.3.7 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fromfp | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.10 | F.10.6.10 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fromfpx | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.11 | F.10.6.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fsub | N/A | |check| | |check| | N/A | |check|\* | N/A | 7.12.14.2 | F.10.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| getpayload | |check| | |check| | |check| | |check| | |check| | |check| | F.10.13.1 | N/A |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| ilogb | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.6.8 | F.10.3.8 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| iscanonical | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.3.2 | N/A |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| issignaling | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.3.8 | N/A |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| ldexp | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.6.9 | F.10.3.9 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| llogb | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.6.10 | F.10.3.10 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| llrint | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.5 | F.10.6.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| llround | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.7 | F.10.6.7 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| logb | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.6.17 | F.10.3.17 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| lrint | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.5 | F.10.6.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| lround | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.7 | F.10.6.7 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| modf | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.6.18 | F.10.3.18 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| nan | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.11.2 | F.10.8.2 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| nearbyint | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.3 | F.10.6.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| nextafter | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.11.3 | F.10.8.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| nextdown | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.11.6 | F.10.8.6 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| nexttoward | |check| | |check| | |check| | |check| | N/A | |check| | 7.12.11.4 | F.10.8.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| nextup | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.11.5 | F.10.8.5 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| remainder | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.10.2 | F.10.7.2 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| remquo | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.10.3 | F.10.7.3 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| rint | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.4 | F.10.6.4 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| round | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.6 | F.10.6.6 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| roundeven | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.8 | F.10.6.8 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| scalbln | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.6.19 | F.10.3.19 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| scalbn | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.6.19 | F.10.3.19 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| setpayload | |check| | |check| | |check| | |check| | |check| | |check| | F.10.13.2 | N/A |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| setpayloadsig | |check| | |check| | |check| | |check| | |check| | |check| | F.10.13.3 | N/A |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| totalorder | |check| | |check| | |check| | |check| | |check| | |check| | F.10.12.1 | N/A |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| totalordermag | |check| | |check| | |check| | |check| | |check| | |check| | F.10.12.2 | N/A |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| trunc | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.9 | F.10.6.9 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| ufromfp | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.10 | F.10.6.10 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| ufromfpx | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.9.11 | F.10.6.11 |
+------------------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
Higher Math Functions
=====================
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+----------++------------+------------------------+----------------------------+
| | (float) | (double) | (long double) | (float16) | (float128) | (bfloat16) | C23 Definition Section | C23 Error Handling Section |
+===========+==================+=================+========================+======================+========================+========================+========================+============================+
| acos | |check| | |check| | | |check| | | | 7.12.4.1 | F.10.1.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| acosh | |check| | | | |check| | | | 7.12.5.1 | F.10.2.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| acospi | | | | |check| | | | 7.12.4.8 | F.10.1.8 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| asin | |check| | |check| | | |check| | | | 7.12.4.2 | F.10.1.2 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| asinh | |check| | | | |check| | | | 7.12.5.2 | F.10.2.2 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| asinpi | | | | |check| | | | 7.12.4.9 | F.10.1.9 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| atan | |check| | 1 ULP | | |check| | | | 7.12.4.3 | F.10.1.3 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| atan2 | |check| | 1 ULP | | | 1 ULP | | 7.12.4.4 | F.10.1.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| atan2pi | | | | | | | 7.12.4.11 | F.10.1.11 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| atanh | |check| | | | |check| | | | 7.12.5.3 | F.10.2.3 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| atanpi | | | | |check| | | | 7.12.4.10 | F.10.1.10 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| cbrt | |check| | |check| | | | | | 7.12.7.1 | F.10.4.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| compoundn | | | | | | | 7.12.7.2 | F.10.4.2 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| cos | |check| | |check| | | |check| | | | 7.12.4.5 | F.10.1.5 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| cosh | |check| | | | |check| | | | 7.12.5.4 | F.10.2.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| cospi | |check| | | | |check| | | | 7.12.4.12 | F.10.1.12 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| dsqrt | N/A | N/A | |check| | N/A | |check|\* | | 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| | | |check| | | | 7.12.6.1 | F.10.3.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| exp10 | |check| | |check| | | |check| | | | 7.12.6.2 | F.10.3.2 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| exp10m1 | |check| | | | |check| | | | 7.12.6.3 | F.10.3.3 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| exp2 | |check| | |check| | | |check| | | | 7.12.6.4 | F.10.3.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| exp2m1 | |check| | | | |check| | | | 7.12.6.5 | F.10.3.5 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| expm1 | |check| | |check| | | |check| | | | 7.12.6.6 | F.10.3.6 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fma | |check| | |check| | | |check| | | | 7.12.13.1 | F.10.10.1 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| f16sqrt | |check|\* | |check|\* | |check|\* | N/A | |check| | | 7.12.14.6 | F.10.11 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| fsqrt | N/A | |check| | |check| | N/A | |check|\* | | 7.12.14.6 | F.10.11 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| hypot | |check| | |check| | | |check| | | | 7.12.7.4 | F.10.4.4 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| lgamma | | | | | | | 7.12.8.3 | F.10.5.3 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| log | |check| | |check| | | |check| | | |check| ? | 7.12.6.11 | F.10.3.11 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| log10 | |check| | |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| | | |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| | 1 ULP | | | | | 7.12.7.5 | F.10.4.5 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| powi\* | | | | | | | | |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| 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 | | | | |check| | | | 7.12.7.9 | F.10.4.9 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| sin | |check| | |check| | | |check| | | | 7.12.4.6 | F.10.1.6 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| sincos | |check| | |check| | | | | | | |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| sinh | |check| | | | |check| | | | 7.12.5.5 | F.10.2.5 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| sinpi | |check| | | | |check| | | | 7.12.4.13 | F.10.1.13 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| sqrt | |check| | |check| | |check| | |check| | |check| | |check| | 7.12.7.10 | F.10.4.10 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| tan | |check| | |check| | | |check| | | | 7.12.4.7 | F.10.1.7 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| tanh | |check| | | | |check| | | | 7.12.5.6 | F.10.2.6 |
+-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+------------------------+----------------------------+
| tanpi | |check| | | | |check| | | | 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.
* \*: LLVM libc extension.
* ? Because of a conflict between float16 logb function and bfloat16 log function, the latter is implemented as `log_bf16`.
..
TODO(lntue): Add a new page to discuss about the algorithms used in the
implementations and include the link here.
GPU Conformance
===============
* Conformance tests are located at: `offload/unittests/Conformance `_.
* The math functions for GPUs are compiled with the following optimization options: ``LIBC_MATH_SKIP_ACCURATE_PASS``, ``LIBC_MATH_INTERMEDIATE_COMP_IN_FLOAT``, ``LIBC_MATH_SMALL_TABLES``, ``LIBC_MATH_NO_ERRNO``, and ``LIBC_MATH_NO_EXCEPT``.
* The conformance test results for higher math functions on GPUs are reported in the table below. The results show the maximum observed ULP distance when comparing a given GPU implementation against the corresponding correctly rounded implementation from LLVM libc, which is computed on the host CPU and serves as the reference. For comparison purposes, results for CUDA Math and HIP Math against the same reference are also included.
+------------------------+-------------+---------------+-----------------------------------------------------------------------------------+
| Function | Test Method | ULP Tolerance | Max ULP Distance |
| | | +--------------------+--------------------+--------------------+--------------------+
| | | | LLVM libc | LLVM libc | CUDA Math | HIP Math |
| | | | (AMDGPU) | (CUDA) | (CUDA) | (AMDGPU) |
+========================+=============+===============+====================+====================+====================+====================+
| acos | Randomized | 4 | 6 (FAILED) | 6 (FAILED) | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acosf | Exhaustive | 4 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acosf16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acoshf | Exhaustive | 4 | 1 | 1 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acoshf16 | Exhaustive | 2 | 0 | 0 | | 0 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| acospif16 | Exhaustive | 2 | 0 | 0 | | |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asin | Randomized | 4 | 6 (FAILED) | 6 (FAILED) | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asinf | Exhaustive | 4 | 1 | 1 | 1 | 3 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asinf16 | Exhaustive | 2 | 0 | 0 | | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asinhf | Exhaustive | 4 | 1 | 1 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| asinhf16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atanf | Exhaustive | 5 | 0 | 0 | 1 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atanf16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atan2f | Randomized | 6 | 1 | 1 | 2 | 3 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atanhf | Exhaustive | 5 | 0 | 0 | 3 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| atanhf16 | Exhaustive | 2 | 0 | 0 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cbrt | Randomized | 2 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cbrtf | Exhaustive | 2 | 0 | 0 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cos | Randomized | 4 | 1 | 1 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cosf | Exhaustive | 4 | 1 | 1 | 2 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cosf16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| coshf | Exhaustive | 4 | 0 | 0 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| coshf16 | Exhaustive | 2 | 1 | 0 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cospif | Exhaustive | 4 | 0 | 0 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| cospif16 | Exhaustive | 2 | 0 | 0 | | |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| erff | Exhaustive | 16 | 0 | 0 | 1 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp | Randomized | 3 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expf | Exhaustive | 3 | 0 | 0 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expf16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp10 | Randomized | 3 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp10f | Exhaustive | 3 | 0 | 0 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp10f16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp2 | Randomized | 3 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp2f | Exhaustive | 3 | 1 | 1 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| exp2f16 | Exhaustive | 2 | 1 | 1 | | 0 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expm1 | Randomized | 3 | 0 | 0 | 1 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expm1f | Exhaustive | 3 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| expm1f16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| hypot | Randomized | 4 | 0 | 0 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| hypotf | Randomized | 4 | 0 | 0 | 1 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| hypotf16 | Exhaustive | 2 | 0 | 0 | | |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log | Randomized | 3 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| logf | Exhaustive | 3 | 1 | 1 | 1 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| logf16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log10 | Randomized | 3 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log10f | Exhaustive | 3 | 1 | 1 | 2 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log10f16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log1p | Randomized | 2 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log1pf | Exhaustive | 2 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log2 | Randomized | 3 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log2f | Exhaustive | 3 | 0 | 0 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| log2f16 | Exhaustive | 2 | 1 | 1 | | 0 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| powf (integer exp.) | Randomized | 16 | 0 | 0 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| powf (real exp.) | Randomized | 16 | 0 | 0 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sin | Randomized | 4 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinf | Exhaustive | 4 | 1 | 1 | 1 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinf16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sincos (cos part) | Randomized | 4 | 1 | 1 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sincos (sin part) | Randomized | 4 | 1 | 1 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sincosf (cos part) | Exhaustive | 4 | 1 | 1 | 2 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sincosf (sin part) | Exhaustive | 4 | 1 | 1 | 1 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinhf | Exhaustive | 4 | 1 | 1 | 3 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinhf16 | Exhaustive | 2 | 1 | 1 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinpif | Exhaustive | 4 | 0 | 0 | 1 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| sinpif16 | Exhaustive | 2 | 0 | 0 | | |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tan | Randomized | 5 | 2 | 2 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanf | Exhaustive | 5 | 0 | 0 | 3 | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanf16 | Exhaustive | 2 | 1 | 1 | | 2 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanhf | Exhaustive | 5 | 0 | 0 | 2 | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanhf16 | Exhaustive | 2 | 0 | 0 | | 1 |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanpif | Exhaustive | 6 | 0 | 0 | | |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
| tanpif16 | Exhaustive | 2 | 1 | 1 | | |
+------------------------+-------------+---------------+--------------------+--------------------+--------------------+--------------------+
Notes:
* Exhaustive tests check every representable point in the input space. This method is used for half-precision functions and single-precision univariate functions.
* Randomized tests check a large, deterministic subset of the input space, typically using 2\ :sup:`32` samples. This method is used for functions with larger input spaces, such as single-precision bivariate and double-precision functions.
* ULP tolerances are based on The Khronos Group, `The OpenCL C Specification v3.0.19 `_, Sec. 7.4, Khronos Registry [July 10, 2025].
* The AMD GPU used for testing is AMD Radeon RX 6950 XT.
* The NVIDIA GPU used for testing is NVIDIA RTX 4000 SFF Ada Generation.
Performance
===========
* Simple performance testings are located at: `libc/test/src/math/performance_testing `_.
* We also use the *perf* tool from the `CORE-MATH `_
project: `link `_.
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 `_
on Linux). Fmod performance results obtained with "performance_testing".
+--------------+-------------------------------+-------------------------------+-------------------------------------+----------------------------------------------------------------------+
| | 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 `_.
* `RLIBM `_.
* `Sollya `_.
* `The CORE-MATH Project `_.
* `The GNU C Library (glibc) `_.
* `The GNU MPFR Library `_.
* `C++23 Standard `_.
* `BFloat16 `_.