| =========================== |
| Math Functions in LLVM-libc |
| =========================== |
| |
| .. role:: raw-html(raw) |
| :format: html |
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| .. |check| replace:: :raw-html:`✅` |
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| .. contents:: Table of Contents |
| :depth: 4 |
| :local: |
| |
| Summary |
| ======= |
| |
| * This document tracks the status of the implementation of math functions in |
| LLVM libc. |
| |
| Implementation Requirements / Goals |
| =================================== |
| |
| * The highest priority is to be as accurate as possible, according to the C and |
| IEEE 754 standards. By default, we will aim to be correctly rounded for `all rounding modes <https://en.cppreference.com/w/c/numeric/fenv/FE_round>`_. |
| The current rounding mode of the floating point environment is used to perform |
| computations and produce the final results. |
| |
| - To test for correctness, we compare the outputs with other correctly rounded |
| multiple-precision math libraries such as the `GNU MPFR library <https://www.mpfr.org/>`_ |
| or the `CORE-MATH library <https://core-math.gitlabpages.inria.fr/>`_. |
| |
| * Our next requirement is that the outputs are consistent across all platforms. |
| Notice that the consistency requirement will be satisfied automatically if the |
| implementation is correctly rounded. |
| |
| * Our last requirement for the implementations is to have good and predicable |
| performance: |
| |
| - The average performance should be comparable to other ``libc`` |
| implementations. |
| - The worst case performance should be within 10X-20X of the average. |
| - Platform-specific implementations or instructions could be added whenever it |
| makes sense and provides significant performance boost. |
| |
| * For other use cases that have strict requirements on the code size, memory |
| footprint, or latency, such as embedded systems, we will aim to be as accurate |
| as possible within the memory or latency budgets, and consistent across all |
| platforms. |
| |
| |
| Source Locations |
| ================ |
| |
| - The main source is located at: `libc/src/math <https://github.com/llvm/llvm-project/tree/main/libc/src/math>`_. |
| - The tests are located at: `libc/test/src/math <https://github.com/llvm/llvm-project/tree/main/libc/test/src/math>`_. |
| - The floating point utilities are located at: `libc/src/__support/FPUtil <https://github.com/llvm/llvm-project/tree/main/libc/src/__support/FPUtil>`_. |
| |
| Add a new math function to LLVM libc |
| ==================================== |
| |
| * To add a new math function, follow the steps at: `libc/src/math/docs/add_math_function.md <https://github.com/llvm/llvm-project/tree/main/libc/src/math/docs/add_math_function.md>`_. |
| |
| Implementation Status |
| ===================== |
| |
| Basic Operations |
| ---------------- |
| |
| ============== ================ =============== ====================== |
| <Func> <Func_f> (float) <Func> (double) <Func_l> (long double) |
| ============== ================ =============== ====================== |
| ceil |check| |check| |check| |
| copysign |check| |check| |check| |
| fabs |check| |check| |check| |
| fdim |check| |check| |check| |
| floor |check| |check| |check| |
| fmax |check| |check| |check| |
| fmin |check| |check| |check| |
| fmod |check| |check| |
| fpclassify |
| frexp |check| |check| |check| |
| ilogb |check| |check| |check| |
| isfinite |
| isgreater |
| isgreaterequal |
| isinf |
| isless |
| islessequal |
| islessgreater |
| isnan |
| isnormal |
| isubordered |
| ldexp |check| |check| |check| |
| llrint |check| |check| |check| |
| llround |check| |check| |check| |
| logb |check| |check| |check| |
| lrint |check| |check| |check| |
| lround |check| |check| |check| |
| modf |check| |check| |check| |
| nan |
| nearbyint |check| |check| |check| |
| nextafter |check| |check| |check| |
| nexttoward |
| remainder |check| |check| |check| |
| remquo |check| |check| |check| |
| rint |check| |check| |check| |
| round |check| |check| |check| |
| scalbn |
| signbit |
| trunc |check| |check| |check| |
| ============== ================ =============== ====================== |
| |
| Higher Math Functions |
| --------------------- |
| |
| ============== ================ =============== ====================== |
| <Func> <Func_f> (float) <Func> (double) <Func_l> (long double) |
| ============== ================ =============== ====================== |
| acos |
| acosh |
| asin |
| asinh |
| atan |
| atan2 |
| atanh |
| cbrt |
| cos |check| |check| |
| cosh |check| |
| erf |
| erfc |
| exp |check| |
| exp2 |check| |
| expm1 |check| |
| fma |check| |check| |
| hypot |check| |check| |
| lgamma |
| log |check| |
| log10 |check| |
| log1p |check| |
| log2 |check| |
| pow |
| sin |check| |check| |
| sincos |check| |check| |
| sinh |check| |
| sqrt |check| |check| |check| |
| tan |
| tanh |check| |
| tgamma |
| ============== ================ =============== ====================== |
| |
| Accuracy of Higher Math Functions |
| ================================= |
| |
| ============== ================ =============== ====================== |
| <Func> <Func_f> (float) <Func> (double) <Func_l> (long double) |
| ============== ================ =============== ====================== |
| cos |check| large |
| cosh |check| |
| exp |check| |
| exp2 |check| |
| expm1 |check| |
| fma |check| |check| |
| hypot |check| |check| |
| log |check| |
| log10 |check| |
| log1p |check| |
| log2 |check| |
| sin |check| large |
| sincos 0.776 ULPs large |
| sinh |check| |
| sqrt |check| |check| |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 (ns) | Latency (ns) | Testing ranges | Testing configuration | |
| | +-----------+-------------------+-----------+-------------------+ +------------+-------------------------+--------------+---------------+ |
| | | LLVM libc | Reference (glibc) | LLVM libc | Reference (glibc) | | CPU | OS | Compiler | Special flags | |
| +==============+===========+===================+===========+===================+=====================================+============+=========================+==============+===============+ |
| | cosf | 14 | 32 | 56 | 59 | :math:`[0, 2\pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| | coshf | 23 | 20 | 73 | 49 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| | expf | 9 | 7 | 44 | 38 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| | exp2f | 8 | 6 | 35 | 23 | :math:`[-10, 10]` | i5-1135G7 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| + +-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| | | 11 | 6 | 49 | 31 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | 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 | 13 | 25 | 57 | 72 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| | log1pf | 16 | 33 | 61 | 97 | :math:`[e^{-0.5} - 1, e^{0.5} - 1]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| | log2f | 13 | 10 | 57 | 46 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| | sinf | 13 | 25 | 54 | 57 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| | sinhf | 23 | 64 | 73 | 141 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| | tanhf | 25 | 59 | 95 | 125 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA | |
| +--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+ |
| |
| 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/>`_. |
