d883a4ad02
Implement sinf function that is correctly rounded to all rounding modes.
- We use a simple range reduction for `pi/16 < |x|` :
Let `k = round(x / pi)` and `y = (x/pi) - k`.
So `k` is an integer and `-0.5 <= y <= 0.5`.
Then
```
sin(x) = sin(y*pi + k*pi)
= (-1)^(k & 1) * sin(y*pi)
~ (-1)^(k & 1) * y * P(y^2)
```
where `y*P(y^2)` is a degree-15 minimax polynomial generated by Sollya with:
```
> P = fpminimax(sin(x*pi)/x, [|0, 2, 4, 6, 8, 10, 12, 14|], [|D...|], [0, 0.5]);
```
- Performance benchmark using perf tool from CORE-MATH project
(https://gitlab.inria.fr/core-math/core-math/-/tree/master) on Ryzen 1700:
Before this patch (not correctly rounded):
```
$ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh sinf
CORE-MATH reciprocal throughput : 17.892
System LIBC reciprocal throughput : 25.559
LIBC reciprocal throughput : 29.381
```
After this patch (correctly rounded):
```
$ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh sinf
CORE-MATH reciprocal throughput : 17.896
System LIBC reciprocal throughput : 25.740
LIBC reciprocal throughput : 27.872
LIBC reciprocal throughput : 20.012 (with `-msse4.2` flag)
LIBC reciprocal throughput : 14.244 (with `-mfma` flag)
```
Reviewed By: zimmermann6
Differential Revision: https://reviews.llvm.org/D123154
238 lines
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ReStructuredText
===========================
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Math Functions in LLVM-libc
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===========================
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.. role:: raw-html(raw)
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:format: html
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.. |check| replace:: :raw-html:`✅`
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.. contents:: Table of Contents
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:depth: 4
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:local:
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Summary
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=======
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* This document tracks the status of the implementation of math functions in
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LLVM libc.
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Implementation Requirements / Goals
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===================================
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* The highest priority is to be as accurate as possible, according to the C and
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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>`_.
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The current rounding mode of the floating point environment is used to perform
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computations and produce the final results.
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- To test for correctness, we compare the outputs with other correctly rounded
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multiple-precision math libraries such as the `GNU MPFR library <https://www.mpfr.org/>`_
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or the `CORE-MATH library <https://core-math.gitlabpages.inria.fr/>`_.
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* Our next requirement is that the outputs are consistent across all platforms.
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Notice that the consistency requirement will be satisfied automatically if the
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implementation is correctly rounded.
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* Our last requirement for the implementations is to have good and predicable
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performance:
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- The average performance should be comparable to other ``libc``
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implementations.
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- The worst case performance should be within 10X-20X of the average.
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- Platform-specific implementations or instructions could be added whenever it
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makes sense and provides significant performance boost.
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* For other use cases that have strict requirements on the code size, memory
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footprint, or latency, such as embedded systems, we will aim to be as accurate
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as possible within the memory or latency budgets, and consistent across all
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platforms.
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Source Locations
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================
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- The main source is located at: `libc/src/math <https://github.com/llvm/llvm-project/tree/main/libc/src/math>`_.
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- The tests are located at: `libc/test/src/math <https://github.com/llvm/llvm-project/tree/main/libc/test/src/math>`_.
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- The floating point utilities are located at: `libc/src/__support/FPUtil <https://github.com/llvm/llvm-project/tree/main/libc/src/__support/FPUtil>`_.
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Add a new math function to LLVM libc
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====================================
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* 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>`_.
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Implementation Status
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=====================
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Basic Operations
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----------------
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============== ================ =============== ======================
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<Func> <Func_f> (float) <Func> (double) <Func_l> (long double)
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============== ================ =============== ======================
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ceil |check| |check| |check|
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copysign |check| |check| |check|
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fabs |check| |check| |check|
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fdim |check| |check| |check|
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floor |check| |check| |check|
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fmax |check| |check| |check|
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fmin |check| |check| |check|
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fmod |check| |check|
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fpclassify
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frexp |check| |check| |check|
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ilogb |check| |check| |check|
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isfinite
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isgreater
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isgreaterequal
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isinf
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isless
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islessequal
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islessgreater
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isnan
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isnormal
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isubordered
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ldexp |check| |check| |check|
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llrint |check| |check| |check|
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llround |check| |check| |check|
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logb |check| |check| |check|
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lrint |check| |check| |check|
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lround |check| |check| |check|
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modf |check| |check| |check|
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nan
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nearbyint |check| |check| |check|
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nextafter |check| |check| |check|
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nexttoward
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remainder |check| |check| |check|
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remquo |check| |check| |check|
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rint |check| |check| |check|
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round |check| |check| |check|
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scalbn
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signbit
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trunc |check| |check| |check|
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============== ================ =============== ======================
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Higher Math Functions
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---------------------
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============== ================ =============== ======================
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<Func> <Func_f> (float) <Func> (double) <Func_l> (long double)
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============== ================ =============== ======================
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acos
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acosh
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asin
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asinh
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atan
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atan2
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atanh
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cbrt
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cos |check| |check|
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cosh
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erf
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erfc
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exp |check|
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exp2 |check|
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expm1 |check|
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fma |check| |check|
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hypot |check| |check|
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lgamma
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log |check|
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log10 |check|
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log1p |check|
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log2 |check|
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pow
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sin |check| |check|
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sincos |check| |check|
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sinh
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sqrt |check| |check| |check|
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tan
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tanh
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tgamma
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============== ================ =============== ======================
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Accuracy of Higher Math Functions
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=================================
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============== ================ =============== ======================
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<Func> <Func_f> (float) <Func> (double) <Func_l> (long double)
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============== ================ =============== ======================
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cos 0.776 ULPs large
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exp |check|
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exp2 |check|
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expm1 |check|
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fma |check| |check|
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hypot |check| |check|
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log |check|
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log10 |check|
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log1p |check|
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log2 |check|
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sin |check| large
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sincos 0.776 ULPs large
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sqrt |check| |check| |check|
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============== ================ =============== ======================
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Legends:
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* |check|: correctly rounded for all 4 rounding modes.
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* CR: correctly rounded for the default rounding mode (round-to-the-nearest,
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tie-to-even).
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* x ULPs: largest errors recorded.
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..
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TODO(lntue): Add a new page to discuss about the algorithms used in the
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implementations and include the link here.
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Performance
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===========
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* 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>`_.
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* We also use the *perf* tool from the `CORE-MATH <https://core-math.gitlabpages.inria.fr/>`_
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project: `link <https://gitlab.inria.fr/core-math/core-math/-/tree/master>`_.
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The performance results from the CORE-MATH's perf tool are reported in the
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table below, using the system library as reference (such as the `GNU C library <https://www.gnu.org/software/libc/>`_
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on Linux). Fmod performance results obtained with "differential_testing".
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+--------------+-------------------------------+-------------------------------+-------------------------------------+---------------------------------------------------------------------+
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| <Func> | Reciprocal throughput (ns) | Latency (ns) | Testing ranges | Testing configuration |
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| +-----------+-------------------+-----------+-------------------+ +------------+-------------------------+--------------+---------------+
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| | LLVM libc | Reference (glibc) | LLVM libc | Reference (glibc) | | CPU | OS | Compiler | Special flags |
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+==============+===========+===================+===========+===================+=====================================+============+=========================+==============+===============+
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| cosf | 37 | 32 | 73 | 72 | :math:`[0, 2\pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| expf | 14 | 9 | 58 | 42 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| exp2f | 25 | 8 | 81 | 37 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| expm1f | 14 | 53 | 59 | 146 | :math:`[-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| fmodf | 73 | 263 | - | - | [MIN_NORMAL, MAX_NORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
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| +-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| | 9 | 11 | - | - | [0, MAX_SUBNORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| fmod | 595 | 3297 | - | - | [MIN_NORMAL, MAX_NORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
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| +-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| | 14 | 13 | - | - | [0, MAX_SUBNORMAL] | i5 mobile | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| hypotf | 25 | 15 | 64 | 49 | :math:`[-10, 10] \times [-10, 10]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| logf | 12 | 10 | 56 | 46 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| log10f | 13 | 25 | 57 | 72 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| 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 |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| log2f | 13 | 10 | 57 | 46 | :math:`[e^{-1}, e]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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| sinf | 14 | 26 | 65 | 59 | :math:`[-\pi, \pi]` | Ryzen 1700 | Ubuntu 20.04 LTS x86_64 | Clang 12.0.0 | FMA |
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+--------------+-----------+-------------------+-----------+-------------------+-------------------------------------+------------+-------------------------+--------------+---------------+
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References
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==========
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* `CRLIBM <https://hal-ens-lyon.archives-ouvertes.fr/ensl-01529804/file/crlibm.pdf>`_.
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* `RLIBM <https://people.cs.rutgers.edu/~sn349/rlibm/>`_.
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* `Sollya <https://www.sollya.org/>`_.
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* `The CORE-MATH Project <https://core-math.gitlabpages.inria.fr/>`_.
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* `The GNU C Library (glibc) <https://www.gnu.org/software/libc/>`_.
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* `The GNU MPFR Library <https://www.mpfr.org/>`_.
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