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Differential D153683

[libc][math] Implement erff function correctly rounded to all rounding modes.
ClosedPublic

Authored by lntue on Jun 23 2023, 9:11 PM.

Details

Reviewers
michaelrj
sivachandra
renyichen
zimmermann6
Commits
rGf320fefc4ad0: [libc][math] Implement erff function correctly rounded to all rounding modes.
Summary

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;

Diff Detail

Event Timeline

lntue created this revision.Jun 23 2023, 9:11 PM
Herald added projects: Restricted Project, Restricted Project. · View Herald TranscriptJun 23 2023, 9:11 PM
Herald added subscribers: luke, frasercrmck, luismarques and 20 others. · View Herald Transcript
lntue requested review of this revision.Jun 23 2023, 9:11 PM
Herald added a subscriber: wangpc. · View Herald TranscriptJun 23 2023, 9:11 PM
lntue added a parent revision: D153682: [libc][math] Clean up exhaustive tests implementations..Jun 23 2023, 9:12 PM
Harbormaster completed remote builds in B240914: Diff 534152.Jun 23 2023, 9:18 PM
lntue edited the summary of this revision. (Show Details)Jun 23 2023, 9:33 PM
Herald added a subscriber: pengfei. · View Herald TranscriptJun 23 2023, 9:34 PM
lntue updated this revision to Diff 534154.Jun 23 2023, 9:41 PM

Fix bazel dependency.

Harbormaster completed remote builds in B240916: Diff 534154.Jun 23 2023, 9:57 PM
lntue updated this revision to Diff 534650.Jun 26 2023, 10:52 AM

Add exhaustive test for erff.

michaelrj added a comment.Jun 27 2023, 4:46 PM

overall LGTM with a nit.

libc/test/src/math/erff_test.cpp
77

does this still need to be here?

lntue updated this revision to Diff 535387.Jun 28 2023, 7:08 AM

Remove unneeded assertion commented out in the unit test.

lntue marked an inline comment as done.Jun 28 2023, 7:08 AM
Harbormaster completed remote builds in B241782: Diff 535387.Jun 28 2023, 7:31 AM
michaelrj accepted this revision.Jun 28 2023, 10:33 AM
This revision is now accepted and ready to land.Jun 28 2023, 10:33 AM
Closed by commit rGf320fefc4ad0: [libc][math] Implement erff function correctly rounded to all rounding modes. (authored by lntue). · Explain WhyJun 28 2023, 10:58 AM
This revision was automatically updated to reflect the committed changes.
lntue added a commit: rGf320fefc4ad0: [libc][math] Implement erff function correctly rounded to all rounding modes..
zimmermann6 added a comment.Jul 4 2023, 4:43 AM

I'm not sure this is related to this ticket, but I cannot compile main any more under Ubuntu:

In file included from /usr/bin/../lib/gcc/x86_64-linux-gnu/13/../../../../include/c++/13/bits/specfun.h:50:
/usr/bin/../lib/gcc/x86_64-linux-gnu/13/../../../../include/c++/13/tr1/ell_integral.tcc:709:14: error: no member named '__throw_domain_error' in namespace 'std'; did you mean '__throw_runtime_error'?
        std::__throw_domain_error(__N("Bad argument in __ellint_3."));
        ~~~~~^
/usr/bin/../lib/gcc/x86_64-linux-gnu/13/../../../../include/c++/13/bits/functexcept.h:131:3: note: '__throw_runtime_error' declared here
  __throw_runtime_error(const char*)
  ^
fatal error: too many errors emitted, stopping now [-ferror-limit=]

It seems clang-14 depends on libstdc++-12-dev, but uses features from libstdc++-13-dev...

lntue added a comment.Jul 4 2023, 10:08 PM
In D153683#4471115, @zimmermann6 wrote:

I'm not sure this is related to this ticket, but I cannot compile main any more under Ubuntu:

In file included from /usr/bin/../lib/gcc/x86_64-linux-gnu/13/../../../../include/c++/13/bits/specfun.h:50:
/usr/bin/../lib/gcc/x86_64-linux-gnu/13/../../../../include/c++/13/tr1/ell_integral.tcc:709:14: error: no member named '__throw_domain_error' in namespace 'std'; did you mean '__throw_runtime_error'?
        std::__throw_domain_error(__N("Bad argument in __ellint_3."));
        ~~~~~^
/usr/bin/../lib/gcc/x86_64-linux-gnu/13/../../../../include/c++/13/bits/functexcept.h:131:3: note: '__throw_runtime_error' declared here
  __throw_runtime_error(const char*)
  ^
fatal error: too many errors emitted, stopping now [-ferror-limit=]

It seems clang-14 depends on libstdc++-12-dev, but uses features from libstdc++-13-dev...

@zimmermann6 :
I don't think it's related to this patch at all, but recently I had problems building with clang on ubuntu also. I found out the root cause for my problems was that after an update,
clang and clang++ versions were mismatched and some dependency were missing. After running apt install / update clang and clang++, the problems went away. You can try to see if it works on your system.

Revision Contents

PathSize
libc/
config/
darwin/
arm/
1 line
linux/
aarch64/
1 line
riscv64/
1 line
x86_64/
1 line
windows/
1 line
docs/
math/
3 lines
spec/
2 lines
src/
math/
2 lines
18 lines
generic/
18 lines
173 lines
test/
src/
math/
13 lines
106 lines
exhaustive/
17 lines
33 lines
utils/
MPFRWrapper/
1 line
8 lines
utils/
bazel/
llvm-project-overlay/
libc/
9 lines

Diff 535387

libc/config/darwin/arm/entrypoints.txt

Show First 20 Lines • Show All 122 Lines • ▼ Show 20 Linesset(TARGET_LIBM_ENTRYPOINTS
libc.src.math.copysign libc.src.math.copysign
libc.src.math.copysignf libc.src.math.copysignf
libc.src.math.copysignl libc.src.math.copysignl
libc.src.math.ceil libc.src.math.ceil
libc.src.math.ceilf libc.src.math.ceilf
libc.src.math.ceill libc.src.math.ceill
libc.src.math.coshf libc.src.math.coshf
libc.src.math.cosf libc.src.math.cosf
libc.src.math.erff
libc.src.math.expf libc.src.math.expf
libc.src.math.exp10f libc.src.math.exp10f
libc.src.math.exp2f libc.src.math.exp2f
libc.src.math.expm1f libc.src.math.expm1f
libc.src.math.fabs libc.src.math.fabs
libc.src.math.fabsf libc.src.math.fabsf
libc.src.math.fabsl libc.src.math.fabsl
libc.src.math.fdim libc.src.math.fdim
▲ Show 20 Lines • Show All 90 Lines • Show Last 20 Lines

libc/config/linux/aarch64/entrypoints.txt

Show First 20 Lines • Show All 234 Lines • ▼ Show 20 Linesset(TARGET_LIBM_ENTRYPOINTS
libc.src.math.copysign libc.src.math.copysign
libc.src.math.copysignf libc.src.math.copysignf
libc.src.math.copysignl libc.src.math.copysignl
libc.src.math.ceil libc.src.math.ceil
libc.src.math.ceilf libc.src.math.ceilf
libc.src.math.ceill libc.src.math.ceill
libc.src.math.coshf libc.src.math.coshf
libc.src.math.cosf libc.src.math.cosf
libc.src.math.erff
libc.src.math.expf libc.src.math.expf
libc.src.math.exp10f libc.src.math.exp10f
libc.src.math.exp2f libc.src.math.exp2f
libc.src.math.expm1f libc.src.math.expm1f
libc.src.math.fabs libc.src.math.fabs
libc.src.math.fabsf libc.src.math.fabsf
libc.src.math.fabsl libc.src.math.fabsl
libc.src.math.fdim libc.src.math.fdim
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libc/config/linux/riscv64/entrypoints.txt

Show First 20 Lines • Show All 241 Lines • ▼ Show 20 Linesset(TARGET_LIBM_ENTRYPOINTS
libc.src.math.copysign libc.src.math.copysign
libc.src.math.copysignf libc.src.math.copysignf
libc.src.math.copysignl libc.src.math.copysignl
libc.src.math.ceil libc.src.math.ceil
libc.src.math.ceilf libc.src.math.ceilf
libc.src.math.ceill libc.src.math.ceill
libc.src.math.coshf libc.src.math.coshf
libc.src.math.cosf libc.src.math.cosf
libc.src.math.erff
libc.src.math.expf libc.src.math.expf
libc.src.math.exp10f libc.src.math.exp10f
libc.src.math.exp2f libc.src.math.exp2f
libc.src.math.expm1f libc.src.math.expm1f
libc.src.math.fabs libc.src.math.fabs
libc.src.math.fabsf libc.src.math.fabsf
libc.src.math.fabsl libc.src.math.fabsl
libc.src.math.fdim libc.src.math.fdim
▲ Show 20 Lines • Show All 273 Lines • Show Last 20 Lines

libc/config/linux/x86_64/entrypoints.txt

Show First 20 Lines • Show All 245 Lines • ▼ Show 20 Linesset(TARGET_LIBM_ENTRYPOINTS
libc.src.math.copysignf libc.src.math.copysignf
libc.src.math.copysignl libc.src.math.copysignl
libc.src.math.ceil libc.src.math.ceil
libc.src.math.ceilf libc.src.math.ceilf
libc.src.math.ceill libc.src.math.ceill
libc.src.math.cos libc.src.math.cos
libc.src.math.coshf libc.src.math.coshf
libc.src.math.cosf libc.src.math.cosf
libc.src.math.erff
libc.src.math.expf libc.src.math.expf
libc.src.math.exp10f libc.src.math.exp10f
libc.src.math.exp2f libc.src.math.exp2f
libc.src.math.expm1f libc.src.math.expm1f
libc.src.math.fabs libc.src.math.fabs
libc.src.math.fabsf libc.src.math.fabsf
libc.src.math.fabsl libc.src.math.fabsl
libc.src.math.fdim libc.src.math.fdim
▲ Show 20 Lines • Show All 274 Lines • Show Last 20 Lines

libc/config/windows/entrypoints.txt

Show First 20 Lines • Show All 121 Lines • ▼ Show 20 Linesset(TARGET_LIBM_ENTRYPOINTS
libc.src.math.copysignf libc.src.math.copysignf
libc.src.math.copysignl libc.src.math.copysignl
libc.src.math.ceil libc.src.math.ceil
libc.src.math.ceilf libc.src.math.ceilf
libc.src.math.ceill libc.src.math.ceill
libc.src.math.cos libc.src.math.cos
libc.src.math.cosf libc.src.math.cosf
libc.src.math.coshf libc.src.math.coshf
libc.src.math.erff
libc.src.math.expf libc.src.math.expf
libc.src.math.exp10f libc.src.math.exp10f
libc.src.math.exp2f libc.src.math.exp2f
libc.src.math.expm1f libc.src.math.expm1f
libc.src.math.fabs libc.src.math.fabs
libc.src.math.fabsf libc.src.math.fabsf
libc.src.math.fabsl libc.src.math.fabsl
libc.src.math.fdim libc.src.math.fdim
▲ Show 20 Lines • Show All 93 Lines • Show Last 20 Lines

libc/docs/math/index.rst

Show First 20 Lines • Show All 336 Lines • ▼ Show 20 Lines
| cosh | | | | | | | | | | | | | | cosh | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| coshf | |check| | |check| | | |check| | |check| | | | |check| | | | | | | coshf | |check| | |check| | | |check| | |check| | | | |check| | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| coshl | | | | | | | | | | | | | | coshl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erf | | | | | | | | | | | | | | erf | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erff | | | | | | | | | | | | | | erff | |check| | |check| | | |check| | |check| | | | |check| | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfl | | | | | | | | | | | | | | erfl | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfc | | | | | | | | | | | | | | erfc | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfcf | | | | | | | | | | | | | | erfcf | | | | | | | | | | | | |
+------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ +------------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+
| erfcl | | | | | | | | | | | | | | erfcl | | | | | | | | | | | | |
▲ Show 20 Lines • Show All 123 Lines • ▼ Show 20 Lines
acos |check| acos |check|
acosh |check| acosh |check|
asin |check| asin |check|
asinh |check| asinh |check|
atan |check| atan |check|
atanh |check| atanh |check|
cos |check| large cos |check| large
cosh |check| cosh |check|
erf |check|
exp |check| exp |check|
exp10 |check| exp10 |check|
exp2 |check| exp2 |check|
expm1 |check| expm1 |check|
fma |check| |check| fma |check| |check|
hypot |check| |check| hypot |check| |check|
log |check| |check| log |check| |check|
log10 |check| |check| log10 |check| |check|
▲ Show 20 Lines • Show All 106 Lines • Show Last 20 Lines

libc/spec/stdc.td

Show First 20 Lines • Show All 426 Lines • ▼ Show 20 Lines HeaderSpec Math = HeaderSpec<
FunctionSpec<"cos", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>, FunctionSpec<"cos", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
FunctionSpec<"cosf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"cosf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"sin", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>, FunctionSpec<"sin", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
FunctionSpec<"sinf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"sinf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"tan", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>, FunctionSpec<"tan", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
FunctionSpec<"tanf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"tanf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"erff", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"expf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"expf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"exp2f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"exp2f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"expm1f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"expm1f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"remainderf", RetValSpec<FloatType>, [ArgSpec<FloatType>, ArgSpec<FloatType>]>, FunctionSpec<"remainderf", RetValSpec<FloatType>, [ArgSpec<FloatType>, ArgSpec<FloatType>]>,
FunctionSpec<"remainder", RetValSpec<DoubleType>, [ArgSpec<DoubleType>, ArgSpec<DoubleType>]>, FunctionSpec<"remainder", RetValSpec<DoubleType>, [ArgSpec<DoubleType>, ArgSpec<DoubleType>]>,
FunctionSpec<"remainderl", RetValSpec<LongDoubleType>, [ArgSpec<LongDoubleType>, ArgSpec<LongDoubleType>]>, FunctionSpec<"remainderl", RetValSpec<LongDoubleType>, [ArgSpec<LongDoubleType>, ArgSpec<LongDoubleType>]>,
▲ Show 20 Lines • Show All 675 Lines • Show Last 20 Lines

libc/src/math/CMakeLists.txt

Show First 20 Lines • Show All 69 Lines • ▼ Show 20 Lines
add_math_entrypoint_object(copysign) add_math_entrypoint_object(copysign)
add_math_entrypoint_object(copysignf) add_math_entrypoint_object(copysignf)
add_math_entrypoint_object(copysignl) add_math_entrypoint_object(copysignl)
add_math_entrypoint_object(cos) add_math_entrypoint_object(cos)
add_math_entrypoint_object(cosf) add_math_entrypoint_object(cosf)
add_math_entrypoint_object(coshf) add_math_entrypoint_object(coshf)
add_math_entrypoint_object(erff)
add_math_entrypoint_object(expf) add_math_entrypoint_object(expf)
add_math_entrypoint_object(exp2f) add_math_entrypoint_object(exp2f)
add_math_entrypoint_object(exp10f) add_math_entrypoint_object(exp10f)
add_math_entrypoint_object(expm1f) add_math_entrypoint_object(expm1f)
▲ Show 20 Lines • Show All 122 Lines • Show Last 20 Lines

libc/src/math/erff.h

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

libc/src/math/generic/CMakeLists.txt

Show First 20 Lines • Show All 525 Lines • ▼ Show 20 Lines HDRS
exp_utils.h exp_utils.h
SRCS SRCS
exp_utils.cpp exp_utils.cpp
DEPENDS DEPENDS
.math_utils .math_utils
) )
add_entrypoint_object( add_entrypoint_object(
erff
SRCS
erff.cpp
HDRS
../erff.h
DEPENDS
.common_constants
libc.src.__support.FPUtil.fp_bits
libc.src.__support.FPUtil.except_value_utils
libc.src.__support.FPUtil.multiply_add
libc.src.__support.FPUtil.polyeval
libc.src.__support.macros.optimization
libc.include.math
COMPILE_OPTIONS
-O3
)
add_entrypoint_object(
expf expf
SRCS SRCS
expf.cpp expf.cpp
HDRS HDRS
../expf.h ../expf.h
DEPENDS DEPENDS
.common_constants .common_constants
libc.src.__support.FPUtil.basic_operations libc.src.__support.FPUtil.basic_operations
▲ Show 20 Lines • Show All 1,022 Lines • Show Last 20 Lines

libc/src/math/generic/erff.cpp

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

libc/test/src/math/CMakeLists.txt

Show First 20 Lines • Show All 1,604 Lines • ▼ Show 20 Lines HDRS
ScalbnTest.h ScalbnTest.h
DEPENDS DEPENDS
libc.include.math libc.include.math
libc.src.math.scalbnl libc.src.math.scalbnl
libc.src.__support.FPUtil.fp_bits libc.src.__support.FPUtil.fp_bits
libc.src.__support.FPUtil.normal_float libc.src.__support.FPUtil.normal_float
) )
add_fp_unittest(
erff_test
NEED_MPFR
SUITE
libc_math_unittests
SRCS
erff_test.cpp
DEPENDS
libc.include.math
libc.src.math.erff
libc.src.__support.FPUtil.fp_bits
)
add_subdirectory(generic) add_subdirectory(generic)
if(NOT LLVM_LIBC_FULL_BUILD) if(NOT LLVM_LIBC_FULL_BUILD)
add_subdirectory(exhaustive) add_subdirectory(exhaustive)
add_subdirectory(differential_testing) add_subdirectory(differential_testing)
endif() endif()

libc/test/src/math/erff_test.cpp

  • This file was added.
//===-- Unittests for erff ------------------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#include "src/__support/FPUtil/FPBits.h"
#include "src/math/erff.h"
#include "test/UnitTest/FPMatcher.h"
#include "test/UnitTest/Test.h"
#include "utils/MPFRWrapper/MPFRUtils.h"
#include <math.h>
#include <errno.h>
#include <stdint.h>
namespace mpfr = __llvm_libc::testing::mpfr;
using __llvm_libc::testing::tlog;
DECLARE_SPECIAL_CONSTANTS(float)
TEST(LlvmLibcErffTest, SpecialNumbers) {
EXPECT_FP_EQ_ALL_ROUNDING(aNaN, __llvm_libc::erff(aNaN));
EXPECT_FP_EQ_ALL_ROUNDING(1.0f, __llvm_libc::erff(inf));
EXPECT_FP_EQ_ALL_ROUNDING(-1.0f, __llvm_libc::erff(neg_inf));
EXPECT_FP_EQ_ALL_ROUNDING(zero, __llvm_libc::erff(zero));
EXPECT_FP_EQ_ALL_ROUNDING(neg_zero, __llvm_libc::erff(neg_zero));
}
TEST(LlvmLibcErffTest, TrickyInputs) {
constexpr int N = 2;
constexpr uint32_t INPUTS[N] = {
0x3f65'9229U, // |x| = 0x1.cb2452p-1f
0x4004'1e6aU, // |x| = 0x1.083cd4p+1f
};
for (int i = 0; i < N; ++i) {
float x = float(FPBits(INPUTS[i]));
EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Erf, x,
__llvm_libc::erff(x), 0.5);
EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Erf, -x,
__llvm_libc::erff(-x), 0.5);
}
}
TEST(LlvmLibcErffTest, InFloatRange) {
constexpr uint32_t COUNT = 234561;
constexpr uint32_t START = 0; // 0
constexpr uint32_t STOP = 0x4080'0000U; // 4.0f
constexpr uint64_t STEP = (STOP - START) / COUNT;
auto test = [&](mpfr::RoundingMode rounding_mode) {
mpfr::ForceRoundingMode __r(rounding_mode);
if (!__r.success)
return;
uint32_t fails = 0;
uint32_t count = 0;
uint32_t cc = 0;
float mx, mr = 0.0;
double tol = 0.5;
for (uint32_t i = 0, v = START; i <= COUNT; ++i, v += STEP) {
float x = FPBits(v).get_val();
if (isnan(x))
continue;
float result = __llvm_libc::erff(x);
++cc;
if (isnan(result))
continue;
++count;
if (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Erf, x, result,
0.5, rounding_mode)) {
michaelrjUnsubmitted
Done
ReplyInline Actions

does this still need to be here?

michaelrj: does this still need to be here?
++fails;
while (!TEST_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Erf, x,
result, tol, rounding_mode)) {
mx = x;
mr = result;
tol *= 2.0;
}
}
}
tlog << " Log failed: " << fails << "/" << count << "/" << cc
<< " tests.\n";
tlog << " Max ULPs is at most: " << static_cast<uint64_t>(tol) << ".\n";
if (fails) {
EXPECT_MPFR_MATCH(mpfr::Operation::Erf, mx, mr, 0.5, rounding_mode);
}
};
tlog << " Test Rounding To Nearest...\n";
test(mpfr::RoundingMode::Nearest);
tlog << " Test Rounding Downward...\n";
test(mpfr::RoundingMode::Downward);
tlog << " Test Rounding Upward...\n";
test(mpfr::RoundingMode::Upward);
tlog << " Test Rounding Toward Zero...\n";
test(mpfr::RoundingMode::TowardZero);
}

libc/test/src/math/exhaustive/CMakeLists.txt

Show First 20 Lines • Show All 86 Lines • ▼ Show 20 Lines DEPENDS
libc.include.math libc.include.math
libc.src.math.tanf libc.src.math.tanf
libc.src.__support.FPUtil.fp_bits libc.src.__support.FPUtil.fp_bits
LINK_LIBRARIES LINK_LIBRARIES
-lpthread -lpthread
) )
add_fp_unittest( add_fp_unittest(
erff_test
NO_RUN_POSTBUILD
NEED_MPFR
SUITE
libc_math_exhaustive_tests
SRCS
erff_test.cpp
DEPENDS
.exhaustive_test
libc.include.math
libc.src.math.erff
libc.src.__support.FPUtil.fp_bits
LINK_LIBRARIES
-lpthread
)
add_fp_unittest(
expf_test expf_test
NO_RUN_POSTBUILD NO_RUN_POSTBUILD
NEED_MPFR NEED_MPFR
SUITE SUITE
libc_math_exhaustive_tests libc_math_exhaustive_tests
SRCS SRCS
expf_test.cpp expf_test.cpp
DEPENDS DEPENDS
▲ Show 20 Lines • Show All 311 Lines • Show Last 20 Lines

libc/test/src/math/exhaustive/erff_test.cpp

  • This file was added.
//===-- Exhaustive test for erff ------------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#include "exhaustive_test.h"
#include "src/math/erff.h"
#include "utils/MPFRWrapper/MPFRUtils.h"
namespace mpfr = __llvm_libc::testing::mpfr;
using LlvmLibcErffExhaustiveTest =
LlvmLibcUnaryOpExhaustiveMathTest<float, mpfr::Operation::Erf,
__llvm_libc::erff>;
// Range: [0, Inf];
static constexpr uint32_t POS_START = 0x0000'0000U;
static constexpr uint32_t POS_STOP = 0x7f80'0000U;
TEST_F(LlvmLibcErffExhaustiveTest, PostiveRange) {
test_full_range_all_roundings(POS_START, POS_STOP);
}
// Range: [-Inf, 0];
static constexpr uint32_t NEG_START = 0xb000'0000U;
static constexpr uint32_t NEG_STOP = 0xff80'0000U;
TEST_F(LlvmLibcErffExhaustiveTest, NegativeRange) {
test_full_range_all_roundings(NEG_START, NEG_STOP);
}

libc/utils/MPFRWrapper/MPFRUtils.h

Show All 28 Linesenum class Operation : int {
Acosh, Acosh,
Asin, Asin,
Asinh, Asinh,
Atan, Atan,
Atanh, Atanh,
Ceil, Ceil,
Cos, Cos,
Cosh, Cosh,
Erf,
Exp, Exp,
Exp2, Exp2,
Exp10, Exp10,
Expm1, Expm1,
Floor, Floor,
Log, Log,
Log2, Log2,
Log10, Log10,
▲ Show 20 Lines • Show All 373 Lines • Show Last 20 Lines

libc/utils/MPFRWrapper/MPFRUtils.cpp

Show First 20 Lines • Show All 223 Lines • ▼ Show 20 Linespublic:
} }
MPFRNumber cosh() const { MPFRNumber cosh() const {
MPFRNumber result(*this); MPFRNumber result(*this);
mpfr_cosh(result.value, value, mpfr_rounding); mpfr_cosh(result.value, value, mpfr_rounding);
return result; return result;
} }
MPFRNumber erf() const {
MPFRNumber result(*this);
mpfr_erf(result.value, value, mpfr_rounding);
return result;
}
MPFRNumber exp() const { MPFRNumber exp() const {
MPFRNumber result(*this); MPFRNumber result(*this);
mpfr_exp(result.value, value, mpfr_rounding); mpfr_exp(result.value, value, mpfr_rounding);
return result; return result;
} }
MPFRNumber exp2() const { MPFRNumber exp2() const {
MPFRNumber result(*this); MPFRNumber result(*this);
▲ Show 20 Lines • Show All 328 Lines • ▼ Show 20 Linesunary_operation(Operation op, InputType input, unsigned int precision,
case Operation::Atanh: case Operation::Atanh:
return mpfrInput.atanh(); return mpfrInput.atanh();
case Operation::Ceil: case Operation::Ceil:
return mpfrInput.ceil(); return mpfrInput.ceil();
case Operation::Cos: case Operation::Cos:
return mpfrInput.cos(); return mpfrInput.cos();
case Operation::Cosh: case Operation::Cosh:
return mpfrInput.cosh(); return mpfrInput.cosh();
case Operation::Erf:
return mpfrInput.erf();
case Operation::Exp: case Operation::Exp:
return mpfrInput.exp(); return mpfrInput.exp();
case Operation::Exp2: case Operation::Exp2:
return mpfrInput.exp2(); return mpfrInput.exp2();
case Operation::Exp10: case Operation::Exp10:
return mpfrInput.exp10(); return mpfrInput.exp10();
case Operation::Expm1: case Operation::Expm1:
return mpfrInput.expm1(); return mpfrInput.expm1();
▲ Show 20 Lines • Show All 446 Lines • Show Last 20 Lines

utils/bazel/llvm-project-overlay/libc/BUILD.bazel

Show First 20 Lines • Show All 1,145 Lines • ▼ Show 20 Lines deps = [
":__support_common", ":__support_common",
":__support_uint128", ":__support_uint128",
":__support_fputil_dyadic_float", ":__support_fputil_dyadic_float",
":common_constants", ":common_constants",
], ],
) )
libc_math_function( libc_math_function(
name = "erff",
additional_deps = [
":__support_fputil_multiply_add",
":__support_fputil_polyeval",
":__support_macros_optimization",
],
)
libc_math_function(
name = "expm1f", name = "expm1f",
additional_deps = [ additional_deps = [
":__support_fputil_fma", ":__support_fputil_fma",
":__support_fputil_multiply_add", ":__support_fputil_multiply_add",
":__support_fputil_nearest_integer", ":__support_fputil_nearest_integer",
":__support_fputil_polyeval", ":__support_fputil_polyeval",
":__support_fputil_rounding_mode", ":__support_fputil_rounding_mode",
":__support_macros_optimization", ":__support_macros_optimization",
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