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[libc][math] Implement erff function correctly rounded to all rounding modes.
Implement correctly rounded `erff` functions. For `x >= 4`, `erff(x) = 1` for `FE_TONEAREST` or `FE_UPWARD`, `0x1.ffffep-1` for `FE_DOWNWARD` or `FE_TOWARDZERO`. For `0 <= x < 4`, we divide into 32 sub-intervals of length `1/8`, and use a degree-15 odd polynomial to approximate `erff(x)` in each sub-interval: ``` erff(x) ~ x * (c0 + c1 * x^2 + c2 * x^4 + ... + c7 * x^14). ``` For `x < 0`, we can use the same formula as above, since the odd part is factored out. Performance tested with `perf.sh` tool from the CORE-MATH project on AMD Ryzen 9 5900X: Reciprocal throughput (clock cycles / op) ``` $ ./perf.sh erff --path2 GNU libc version: 2.35 GNU libc release: stable -- CORE-MATH reciprocal throughput -- with -march=native (with FMA instructions) [####################] 100 % Ntrial = 20 ; Min = 11.790 + 0.182 clc/call; Median-Min = 0.154 clc/call; Max = 12.255 clc/call; -- CORE-MATH reciprocal throughput -- with -march=x86-64-v2 (without FMA instructions) [####################] 100 % Ntrial = 20 ; Min = 14.205 + 0.151 clc/call; Median-Min = 0.159 clc/call; Max = 15.893 clc/call; -- System LIBC reciprocal throughput -- [####################] 100 % Ntrial = 20 ; Min = 45.519 + 0.445 clc/call; Median-Min = 0.552 clc/call; Max = 46.345 clc/call; -- LIBC reciprocal throughput -- with -mavx2 -mfma (with FMA instructions) [####################] 100 % Ntrial = 20 ; Min = 9.595 + 0.214 clc/call; Median-Min = 0.220 clc/call; Max = 9.887 clc/call; -- LIBC reciprocal throughput -- with -msse4.2 (without FMA instructions) [####################] 100 % Ntrial = 20 ; Min = 10.223 + 0.190 clc/call; Median-Min = 0.222 clc/call; Max = 10.474 clc/call; ``` and latency (clock cycles / op): ``` $ ./perf.sh erff --path2 GNU libc version: 2.35 GNU libc release: stable -- CORE-MATH latency -- with -march=native (with FMA instructions) [####################] 100 % Ntrial = 20 ; Min = 38.566 + 0.391 clc/call; Median-Min = 0.503 clc/call; Max = 39.170 clc/call; -- CORE-MATH latency -- with -march=x86-64-v2 (without FMA instructions) [####################] 100 % Ntrial = 20 ; Min = 43.223 + 0.667 clc/call; Median-Min = 0.680 clc/call; Max = 43.913 clc/call; -- System LIBC latency -- [####################] 100 % Ntrial = 20 ; Min = 111.613 + 1.267 clc/call; Median-Min = 1.696 clc/call; Max = 113.444 clc/call; -- LIBC latency -- with -mavx2 -mfma (with FMA instructions) [####################] 100 % Ntrial = 20 ; Min = 40.138 + 0.410 clc/call; Median-Min = 0.536 clc/call; Max = 40.729 clc/call; -- LIBC latency -- with -msse4.2 (without FMA instructions) [####################] 100 % Ntrial = 20 ; Min = 44.858 + 0.872 clc/call; Median-Min = 0.814 clc/call; Max = 46.019 clc/call; ``` Reviewed By: michaelrj Differential Revision: https://reviews.llvm.org/D153683
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‎libc/config/darwin/arm/entrypoints.txt

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@@ -128,6 +128,7 @@ set(TARGET_LIBM_ENTRYPOINTS
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libc.src.math.ceill
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libc.src.math.coshf
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libc.src.math.cosf
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libc.src.math.erff
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libc.src.math.expf
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libc.src.math.exp10f
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libc.src.math.exp2f

‎libc/config/linux/aarch64/entrypoints.txt

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@@ -240,6 +240,7 @@ set(TARGET_LIBM_ENTRYPOINTS
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libc.src.math.ceill
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libc.src.math.coshf
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libc.src.math.cosf
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libc.src.math.erff
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libc.src.math.expf
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libc.src.math.exp10f
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libc.src.math.exp2f

‎libc/config/linux/riscv64/entrypoints.txt

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@@ -247,6 +247,7 @@ set(TARGET_LIBM_ENTRYPOINTS
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libc.src.math.ceill
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libc.src.math.coshf
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libc.src.math.cosf
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libc.src.math.erff
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libc.src.math.expf
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libc.src.math.exp10f
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libc.src.math.exp2f

‎libc/config/linux/x86_64/entrypoints.txt

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libc.src.math.cos
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libc.src.math.coshf
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libc.src.math.cosf
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libc.src.math.erff
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libc.src.math.expf
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libc.src.math.exp10f
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libc.src.math.exp2f

‎libc/config/windows/entrypoints.txt

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@@ -127,6 +127,7 @@ set(TARGET_LIBM_ENTRYPOINTS
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libc.src.math.cos
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libc.src.math.cosf
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libc.src.math.coshf
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libc.src.math.erff
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libc.src.math.expf
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libc.src.math.exp10f
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libc.src.math.exp2f

‎libc/docs/math/index.rst

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@@ -342,7 +342,7 @@ Higher Math Functions
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+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
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| erf | | | | | | | | | | | | |
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+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
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| erff | | | | | | | | | | | | |
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| erff | |check| | |check| | | |check| | |check| | | | |check| | | | | |
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+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
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| erfl | | | | | | | | | | | | |
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+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
@@ -482,6 +482,7 @@ atan |check|
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atanh |check|
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cos |check| large
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cosh |check|
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erf |check|
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exp |check|
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exp10 |check|
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exp2 |check|

‎libc/spec/stdc.td

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@@ -432,6 +432,8 @@ def StdC : StandardSpec<"stdc"> {
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FunctionSpec<"tan", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
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FunctionSpec<"tanf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
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FunctionSpec<"erff", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
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FunctionSpec<"expf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
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FunctionSpec<"exp2f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
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FunctionSpec<"expm1f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,

‎libc/src/math/CMakeLists.txt

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@@ -75,6 +75,8 @@ add_math_entrypoint_object(cos)
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add_math_entrypoint_object(cosf)
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add_math_entrypoint_object(coshf)
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add_math_entrypoint_object(erff)
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add_math_entrypoint_object(expf)
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add_math_entrypoint_object(exp2f)

‎libc/src/math/erff.h

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//===-- Implementation header for erff --------------------------*- C++ -*-===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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#ifndef LLVM_LIBC_SRC_MATH_ERFF_H
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#define LLVM_LIBC_SRC_MATH_ERFF_H
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namespace __llvm_libc {
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float erff(float x);
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} // namespace __llvm_libc
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#endif // LLVM_LIBC_SRC_MATH_ERFF_H

‎libc/src/math/generic/CMakeLists.txt

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@@ -530,6 +530,24 @@ add_object_library(
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.math_utils
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)
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add_entrypoint_object(
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erff
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SRCS
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erff.cpp
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HDRS
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../erff.h
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DEPENDS
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.common_constants
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libc.src.__support.FPUtil.fp_bits
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libc.src.__support.FPUtil.except_value_utils
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libc.src.__support.FPUtil.multiply_add
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libc.src.__support.FPUtil.polyeval
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libc.src.__support.macros.optimization
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libc.include.math
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COMPILE_OPTIONS
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-O3
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)
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add_entrypoint_object(
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expf
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SRCS

‎libc/src/math/generic/erff.cpp

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//===-- Single-precision erf(x) function ----------------------------------===//
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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//
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//===----------------------------------------------------------------------===//
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#include "src/math/erff.h"
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#include "src/__support/FPUtil/FPBits.h"
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#include "src/__support/FPUtil/PolyEval.h"
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#include "src/__support/FPUtil/except_value_utils.h"
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#include "src/__support/FPUtil/multiply_add.h"
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#include "src/__support/common.h"
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#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
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namespace __llvm_libc {
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// Polynomials approximating erf(x)/x on ( k/8, (k + 1)/8 ) generated by Sollya
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// with:
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// > P = fpminimax(erf(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14|], [|D...|],
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// [k/8, (k + 1)/8]);
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// for k = 0..31.
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constexpr double COEFFS[32][8] = {
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{0x1.20dd750429b6dp0, -0x1.812746b037753p-2, 0x1.ce2f219e8596ap-4,
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-0x1.b82cdacb78fdap-6, 0x1.56479297dfda5p-8, -0x1.8b3ac5455ef02p-11,
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-0x1.126fcac367e3bp-8, 0x1.2d0bdb3ba4984p-4},
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{0x1.20dd750429b6dp0, -0x1.812746b0379a8p-2, 0x1.ce2f21a03cf2ap-4,
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-0x1.b82ce30de083ep-6, 0x1.565bcad3eb60fp-8, -0x1.c02c66f659256p-11,
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0x1.f92f673385229p-14, -0x1.def402648ae9p-17},
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{0x1.20dd750429b34p0, -0x1.812746b032dcep-2, 0x1.ce2f219d84aaep-4,
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-0x1.b82ce22dcf139p-6, 0x1.565b9efcd4af1p-8, -0x1.c021f1af414bcp-11,
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0x1.f7c6d177eff82p-14, -0x1.c9e4410dcf865p-17},
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{0x1.20dd750426eabp0, -0x1.812746ae592c7p-2, 0x1.ce2f211525f14p-4,
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-0x1.b82ccc125e63fp-6, 0x1.56596f261cfd3p-8, -0x1.bfde1ff8eeecfp-11,
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0x1.f31a9d15dc5d8p-14, -0x1.a5a4362844b3cp-17},
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{0x1.20dd75039c705p0, -0x1.812746777e74dp-2, 0x1.ce2f17af98a1bp-4,
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-0x1.b82be4b817cbep-6, 0x1.564bec2e2962ep-8, -0x1.bee86f9da3558p-11,
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0x1.e9443689dc0ccp-14, -0x1.79c0f230805d8p-17},
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{0x1.20dd74f811211p0, -0x1.81274371a3e8fp-2, 0x1.ce2ec038262e5p-4,
41+
-0x1.b8265b82c5e1fp-6, 0x1.5615a2e239267p-8, -0x1.bc63ae023dcebp-11,
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0x1.d87c2102f7e06p-14, -0x1.49584bea41d62p-17},
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{0x1.20dd746d063e3p0, -0x1.812729a8a950fp-2, 0x1.ce2cb0a2df232p-4,
44+
-0x1.b80eca1f51278p-6, 0x1.5572e26c46815p-8, -0x1.b715e5638b65ep-11,
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0x1.bfbb195484968p-14, -0x1.177a565c15c52p-17},
46+
{0x1.20dd701b44486p0, -0x1.812691145f237p-2, 0x1.ce23a06b8cfd9p-4,
47+
-0x1.b7c1dc7245288p-6, 0x1.53e92f7f397ddp-8, -0x1.ad97cc4acf0b2p-11,
48+
0x1.9f028b2b09b71p-14, -0x1.cdc4da08da8c1p-18},
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{0x1.20dd5715ac332p0, -0x1.8123e680bd0ebp-2, 0x1.ce0457aded691p-4,
50+
-0x1.b6f52d52bed4p-6, 0x1.50c291b84414cp-8, -0x1.9ea246b1ad4a9p-11,
51+
0x1.77654674e0cap-14, -0x1.737c11a1bcebbp-18},
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{0x1.20dce6593e114p0, -0x1.811a59c02eadcp-2, 0x1.cdab53c7cd7d5p-4,
53+
-0x1.b526d2e321eedp-6, 0x1.4b1d32cd8b994p-8, -0x1.8963143ec0a1ep-11,
54+
0x1.4ad5700e4db91p-14, -0x1.231e100e43ef2p-18},
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{0x1.20db48bfd5a62p0, -0x1.80fdd84f9e308p-2, 0x1.ccd340d462983p-4,
56+
-0x1.b196a2928768p-6, 0x1.4210c2c13a0f7p-8, -0x1.6dbdfb4ff71aep-11,
57+
0x1.1bca2d17fbd71p-14, -0x1.bca36f90c7cf5p-19},
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{0x1.20d64b2f8f508p0, -0x1.80b4d4f19fa8bp-2, 0x1.cb088197262e3p-4,
59+
-0x1.ab51fd02e5b99p-6, 0x1.34e1e5e81a632p-8, -0x1.4c66377b502cep-11,
60+
0x1.d9ad25066213cp-15, -0x1.4b0df7dd0cfa1p-19},
61+
{0x1.20c8fc1243576p0, -0x1.8010cb2009e27p-2, 0x1.c7a47e9299315p-4,
62+
-0x1.a155be5683654p-6, 0x1.233502694997bp-8, -0x1.26c94b7d813p-11,
63+
0x1.8094f1de25fb9p-15, -0x1.e0e3d776c6eefp-20},
64+
{0x1.20a9bd1611bc1p0, -0x1.7ec7fbce83f9p-2, 0x1.c1d757d7317b7p-4,
65+
-0x1.92c160cd589fp-6, 0x1.0d307269cc5c2p-8, -0x1.fda5b0d2d1879p-12,
66+
0x1.2fdd7b3b14a7fp-15, -0x1.54eed4a26af5ap-20},
67+
{0x1.20682834f943dp0, -0x1.7c73f747bf5a9p-2, 0x1.b8c2db4a9ffd1p-4,
68+
-0x1.7f0e4ffe989ecp-6, 0x1.e7061eae4166ep-9, -0x1.ad36e873fff2dp-12,
69+
0x1.d39222396128ep-16, -0x1.d83dacec5ea6bp-21},
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{0x1.1feb8d12676d7p0, -0x1.7898347284afep-2, 0x1.aba3466b34451p-4,
71+
-0x1.663adc573e2f9p-6, 0x1.ae99fb17c3e08p-9, -0x1.602f950ad5535p-12,
72+
0x1.5e9717490609dp-16, -0x1.3fca107bbc8d5p-21},
73+
{0x1.1f12fe3c536fap0, -0x1.72b1d1f22e6d3p-2, 0x1.99fc0eed4a896p-4,
74+
-0x1.48db0a87bd8c6p-6, 0x1.73e368895aa61p-9, -0x1.19b35d5301fc8p-12,
75+
0x1.007987e4bb033p-16, -0x1.a7edcd4c2dc7p-22},
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{0x1.1db7b0df84d5dp0, -0x1.6a4e4a41cde02p-2, 0x1.83bbded16455dp-4,
77+
-0x1.2809b3b36977ep-6, 0x1.39c08bab44679p-9, -0x1.b7b45a70ed119p-13,
78+
0x1.6e99b36410e7bp-17, -0x1.13619bb7ebc0cp-22},
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{0x1.1bb1c85c4a527p0, -0x1.5f23b99a249a3p-2, 0x1.694c91fa0d12cp-4,
80+
-0x1.053e1ce11c72dp-6, 0x1.02bf72c50ea78p-9, -0x1.4f478fb56cb02p-13,
81+
0x1.005f80ecbe213p-17, -0x1.5f2446bde7f5bp-23},
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{0x1.18dec3bd51f9dp0, -0x1.5123f58346186p-2, 0x1.4b8a1ca536ab4p-4,
83+
-0x1.c4243015cc723p-7, 0x1.a1a8a01d351efp-10, -0x1.f466b34f1d86bp-14,
84+
0x1.5f835eea0bf6ap-18, -0x1.b83165b939234p-24},
85+
{0x1.152804c3369f4p0, -0x1.4084cd4afd4bcp-2, 0x1.2ba2e836e47aap-4,
86+
-0x1.800f2dfc6904bp-7, 0x1.4a6daf0669c59p-10, -0x1.6e326ab872317p-14,
87+
0x1.d9761a6a755a5p-19, -0x1.0fca33f9dd4b5p-24},
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{0x1.1087ad68356aap0, -0x1.2dbb044707459p-2, 0x1.0aea8ceaa0384p-4,
89+
-0x1.40b516d52b3d2p-7, 0x1.00c9e05f01d22p-10, -0x1.076afb0dc0ff7p-14,
90+
0x1.39fadec400657p-19, -0x1.4b5761352e7e3p-25},
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{0x1.0b0a7a8ba4a22p0, -0x1.196990d22d4a1p-2, 0x1.d5551e6ac0c4dp-5,
92+
-0x1.07cce1770bd1ap-7, 0x1.890347b8848bfp-11, -0x1.757ec96750b6ap-15,
93+
0x1.9b258a1e06bcep-20, -0x1.8fc6d22da7572p-26},
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{0x1.04ce2be70fb47p0, -0x1.0449e4b0b9cacp-2, 0x1.97f7424f4b0e7p-5,
95+
-0x1.ac825439c42f4p-8, 0x1.28f5f65426dfbp-11, -0x1.05b699a90f90fp-15,
96+
0x1.0a888eecf4593p-20, -0x1.deace2b32bb31p-27},
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{0x1.fbf9fb0e11cc8p-1, -0x1.de2640856545ap-3, 0x1.5f5b1f47f851p-5,
98+
-0x1.588bc71eb41b9p-8, 0x1.bc6a0a772f56dp-12, -0x1.6b9fad1f1657ap-16,
99+
0x1.573204ba66504p-21, -0x1.1d38065c94e44p-27},
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{0x1.ed8f18c99e031p-1, -0x1.b4cb6acd903b4p-3, 0x1.2c7f3dddd6fc1p-5,
101+
-0x1.13052067df4ep-8, 0x1.4a5027444082fp-12, -0x1.f672bab0e2554p-17,
102+
0x1.b83c756348cc9p-22, -0x1.534f1a1079499p-28},
103+
{0x1.debd33044166dp-1, -0x1.8d7cd9053f7d8p-3, 0x1.ff9957fb3d6e7p-6,
104+
-0x1.b50be55de0f36p-9, 0x1.e92c8ec53a628p-13, -0x1.5a4b88d508007p-17,
105+
0x1.1a27737559e26p-22, -0x1.942ae62cb2c14p-29},
106+
{0x1.cfdbf0386f3bdp-1, -0x1.68e33d93b0dc4p-3, 0x1.b2683d58f53dep-6,
107+
-0x1.5a9174e70d26fp-9, 0x1.69ddd326d49cdp-13, -0x1.dd8f397a8219cp-18,
108+
0x1.6a755016ad4ddp-23, -0x1.e366e0139187dp-30},
109+
{0x1.c132adb8d7464p-1, -0x1.475a899f61b46p-3, 0x1.70a431397a77cp-6,
110+
-0x1.12e3d35beeee2p-9, 0x1.0c16b05738333p-13, -0x1.4a47f873e144ep-18,
111+
0x1.d3d494c698c02p-24, -0x1.2302c59547fe5p-30},
112+
{0x1.b2f5fd05555e7p-1, -0x1.28feefbe03ec7p-3, 0x1.3923acbb3a676p-6,
113+
-0x1.b4ff793cd6358p-10, 0x1.8ea0eb8c913bcp-14, -0x1.cb31ec2baceb1p-19,
114+
0x1.30011e7e80c04p-24, -0x1.617710635cb1dp-31},
115+
{0x1.a54853cd9593ep-1, -0x1.0dbdbaea4dc8ep-3, 0x1.0a93e2c20a0fdp-6,
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-0x1.5c969ff401ea8p-10, 0x1.29e0cc64fe627p-14, -0x1.4160d8e9d3c2ap-19,
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0x1.8e7b67594624ap-25, -0x1.b1cf2c975b09bp-32},
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{0x1.983ceece09ff8p-1, -0x1.eacc78f7a2dp-4, 0x1.c74418410655fp-7,
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-0x1.1756a050e441ep-10, 0x1.bff3650f7f548p-15, -0x1.c56c0217d3adap-20,
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0x1.07b4918d0b489p-25, -0x1.0d4be8c1c50f8p-32},
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};
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LLVM_LIBC_FUNCTION(float, erff, (float x)) {
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using FPBits = typename fputil::FPBits<float>;
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FPBits xbits(x);
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uint32_t x_u = xbits.uintval();
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uint32_t x_abs = x_u & 0x7fff'ffffU;
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// Exceptional values
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if (LIBC_UNLIKELY(x_abs == 0x3f65'9229U)) // |x| = 0x1.cb2452p-1f
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return x < 0.0f ? fputil::round_result_slightly_down(-0x1.972ea8p-1f)
133+
: fputil::round_result_slightly_up(0x1.972ea8p-1f);
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if (LIBC_UNLIKELY(x_abs == 0x4004'1e6aU)) // |x| = 0x1.083cd4p+1f
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return x < 0.0f ? fputil::round_result_slightly_down(-0x1.fe3462p-1f)
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: fputil::round_result_slightly_up(0x1.fe3462p-1f);
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// if (LIBC_UNLIKELY(x_abs > 0x407a'd444U)) {
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if (LIBC_UNLIKELY(x_abs >= 0x4080'0000U)) {
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const float ONE[2] = {1.0f, -1.0f};
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const float SMALL[2] = {-0x1.0p-25f, 0x1.0p-25f};
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int sign = static_cast<int>(xbits.get_sign());
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if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) {
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return (x_abs > 0x7f80'0000) ? x : ONE[sign];
147+
}
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return ONE[sign] + SMALL[sign];
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}
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// Polynomial approximation:
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// erf(x) ~ x * (c0 + c1 * x^2 + c2 * x^4 + ... + c7 * x^14)
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double xd = static_cast<double>(x);
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double xsq = xd * xd;
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const uint32_t EIGHT = 3 << FPBits::FloatProp::MANTISSA_WIDTH;
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int idx = static_cast<int>(FPBits(x_abs + EIGHT).get_val());
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double x4 = xsq * xsq;
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double c0 = fputil::multiply_add(xsq, COEFFS[idx][1], COEFFS[idx][0]);
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double c1 = fputil::multiply_add(xsq, COEFFS[idx][3], COEFFS[idx][2]);
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double c2 = fputil::multiply_add(xsq, COEFFS[idx][5], COEFFS[idx][4]);
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double c3 = fputil::multiply_add(xsq, COEFFS[idx][7], COEFFS[idx][6]);
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double x8 = x4 * x4;
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double p0 = fputil::multiply_add(x4, c1, c0);
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double p1 = fputil::multiply_add(x4, c3, c2);
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return static_cast<float>(xd * fputil::multiply_add(x8, p1, p0));
171+
}
172+
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} // namespace __llvm_libc

‎libc/test/src/math/CMakeLists.txt

Lines changed: 13 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -1610,6 +1610,19 @@ add_fp_unittest(
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libc.src.__support.FPUtil.normal_float
16111611
)
16121612

1613+
add_fp_unittest(
1614+
erff_test
1615+
NEED_MPFR
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SUITE
1617+
libc_math_unittests
1618+
SRCS
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erff_test.cpp
1620+
DEPENDS
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libc.include.math
1622+
libc.src.math.erff
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libc.src.__support.FPUtil.fp_bits
1624+
)
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16131626
add_subdirectory(generic)
16141627

16151628
if(NOT LLVM_LIBC_FULL_BUILD)

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