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Differential D150374

[libc][math] Implement double precision log2 function correctly rounded to all rounding modes.
ClosedPublic

Authored by lntue on May 11 2023, 8:11 AM.

Details

Reviewers
michaelrj
sivachandra
renyichen
santoshn
zimmermann6
cqlauter
Commits
rG111d27484132: [libc][math] Implement double precision log2 function correctly rounded to all…
Summary

Implement double precision log2 function correctly rounded to all
rounding modes.

See https://reviews.llvm.org/D150014 for a more detail description of the algorithm.

Performance

  • For 0.5 <= x <= 2, the fast pass hitting rate is about 99.91%.
  • Reciprocal throughput from CORE-MATH's perf tool on Ryzen 5900X:
$ ./perf.sh log2
GNU libc version: 2.35
GNU libc release: stable

-- CORE-MATH reciprocal throughput -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 15.458 + 0.204 clc/call; Median-Min = 0.224 clc/call; Max = 15.867 clc/call;

-- CORE-MATH reciprocal throughput -- without FMA (-march=x86-64-v2)
[####################] 100 %
Ntrial = 20 ; Min = 23.711 + 0.524 clc/call; Median-Min = 0.443 clc/call; Max = 25.307 clc/call;

-- System LIBC reciprocal throughput --
[####################] 100 %
Ntrial = 20 ; Min = 14.807 + 0.199 clc/call; Median-Min = 0.211 clc/call; Max = 15.137 clc/call;

-- LIBC reciprocal throughput -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 17.666 + 0.274 clc/call; Median-Min = 0.298 clc/call; Max = 18.531 clc/call;

-- LIBC reciprocal throughput -- without FMA
[####################] 100 %
Ntrial = 20 ; Min = 26.534 + 0.418 clc/call; Median-Min = 0.462 clc/call; Max = 27.327 clc/call;
  • Latency from CORE-MATH's perf tool on Ryzen 5900X:
$ ./perf.sh log2 --latency
GNU libc version: 2.35
GNU libc release: stable

-- CORE-MATH latency -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 46.048 + 1.643 clc/call; Median-Min = 1.694 clc/call; Max = 48.018 clc/call;

-- CORE-MATH latency -- without FMA (-march=x86-64-v2)
[####################] 100 %
Ntrial = 20 ; Min = 62.333 + 0.138 clc/call; Median-Min = 0.119 clc/call; Max = 62.583 clc/call;

-- System LIBC latency --
[####################] 100 %
Ntrial = 20 ; Min = 45.206 + 1.503 clc/call; Median-Min = 1.467 clc/call; Max = 47.229 clc/call;

-- LIBC latency -- with FMA
[####################] 100 %
Ntrial = 20 ; Min = 43.042 + 0.454 clc/call; Median-Min = 0.484 clc/call; Max = 43.912 clc/call;

-- LIBC latency -- without FMA
[####################] 100 %
Ntrial = 20 ; Min = 57.016 + 1.636 clc/call; Median-Min = 1.655 clc/call; Max = 58.816 clc/call;
  • Accurate pass latency:
$ ./perf.sh log2 --latency --simple_stat
GNU libc version: 2.35
GNU libc release: stable

-- CORE-MATH latency -- with FMA
177.632

-- CORE-MATH latency -- without FMA (-march=x86-64-v2)
231.332

-- LIBC latency -- with FMA
459.751

-- LIBC latency -- without FMA
463.850

Diff Detail

Event Timeline

lntue created this revision.May 11 2023, 8:11 AM
Herald added projects: Restricted Project, Restricted Project. · View Herald TranscriptMay 11 2023, 8:11 AM
Herald added subscribers: ecnelises, tschuett. · View Herald Transcript
lntue requested review of this revision.May 11 2023, 8:11 AM
Harbormaster completed remote builds in B231343: Diff 521318.May 11 2023, 8:12 AM
lntue edited the summary of this revision. (Show Details)May 11 2023, 8:27 AM
Herald added a subscriber: pengfei. · View Herald TranscriptMay 11 2023, 8:27 AM
lntue added a parent revision: D150131: [libc][math] Implement double precision log function correctly rounded to all rounding modes..May 11 2023, 8:28 AM
lntue updated this revision to Diff 524105.May 21 2023, 7:54 AM

Sync to HEAD.

Harbormaster completed remote builds in B233429: Diff 524105.May 21 2023, 8:01 AM
lntue updated this revision to Diff 524121.May 21 2023, 11:30 AM

Changed log2 function name in memory_utils.

Harbormaster completed remote builds in B233443: Diff 524121.May 21 2023, 11:36 AM
lntue added a child revision: D151049: [libc][math] Implement double precision log1p correctly rounded to all rounding modes..May 21 2023, 11:44 AM
lntue updated this revision to Diff 524149.May 21 2023, 8:00 PM

Update utils_test.cpp

Harbormaster completed remote builds in B233464: Diff 524149.May 21 2023, 8:06 PM
zimmermann6 requested changes to this revision.May 23 2023, 6:33 AM

I get an error while running ninja:

CMake Error at /localdisk/zimmerma/llvm-project/libc/cmake/modules/LLVMLibCLibraryRules.cmake:5 (get_target_property):
  get_target_property() called with non-existent target
  "libc.src.math.generic.log_range_reduction".
Call Stack (most recent call first):
  /localdisk/zimmerma/llvm-project/libc/cmake/modules/LLVMLibCLibraryRules.cmake:35 (collect_object_file_deps)
  /localdisk/zimmerma/llvm-project/libc/cmake/modules/LLVMLibCLibraryRules.cmake:35 (collect_object_file_deps)
  /localdisk/zimmerma/llvm-project/libc/cmake/modules/LLVMLibCLibraryRules.cmake:82 (collect_object_file_deps)
  /localdisk/zimmerma/llvm-project/libc/lib/CMakeLists.txt:26 (add_entrypoint_library)
This revision now requires changes to proceed.May 23 2023, 6:33 AM
lntue updated this revision to Diff 524708.May 23 2023, 7:36 AM

Sync to HEAD.

zimmermann6 accepted this revision.May 23 2023, 7:44 AM

here is what I get on my AMD EPYC 7282 for the reciprocal throughput:

GNU libc version: 2.36
GNU libc release: stable
[####################] 100 %
Ntrial = 20 ; Min = 21.110 + 0.272 clc/call; Median-Min = 0.280 clc/call; Max = 21.622 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 21.312 + 0.198 clc/call; Median-Min = 0.085 clc/call; Max = 23.254 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 31.690 + 0.395 clc/call; Median-Min = 0.345 clc/call; Max = 34.334 clc/call;

and for the latency:

GNU libc version: 2.36
GNU libc release: stable
[####################] 100 %
Ntrial = 20 ; Min = 58.310 + 0.416 clc/call; Median-Min = 0.304 clc/call; Max = 59.411 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 55.688 + 0.187 clc/call; Median-Min = 0.093 clc/call; Max = 56.306 clc/call;
[####################] 100 %
Ntrial = 20 ; Min = 62.254 + 0.355 clc/call; Median-Min = 0.339 clc/call; Max = 62.856 clc/call;
This revision is now accepted and ready to land.May 23 2023, 7:44 AM
This revision was landed with ongoing or failed builds.May 23 2023, 7:49 AM
Closed by commit rG111d27484132: [libc][math] Implement double precision log2 function correctly rounded to all… (authored by lntue). · Explain Why
This revision was automatically updated to reflect the committed changes.
lntue added a commit: rG111d27484132: [libc][math] Implement double precision log2 function correctly rounded to all….
Harbormaster completed remote builds in B233866: Diff 524708.May 23 2023, 7:59 AM

Revision Contents

PathSize
libc/
config/
darwin/
arm/
1 line
linux/
aarch64/
1 line
x86_64/
1 line
windows/
1 line
spec/
1 line
src/
math/
1 line
generic/
20 lines
957 lines
18 lines
string/
memory_utils/
8 lines
test/
src/
math/
14 lines
138 lines

Diff 524121

libc/config/darwin/arm/entrypoints.txt

Show First 20 Lines • Show All 171 Lines • ▼ Show 20 Linesset(TARGET_LIBM_ENTRYPOINTS
libc.src.math.lroundf libc.src.math.lroundf
libc.src.math.lroundl libc.src.math.lroundl
libc.src.math.ldexp libc.src.math.ldexp
libc.src.math.ldexpf libc.src.math.ldexpf
libc.src.math.ldexpl libc.src.math.ldexpl
libc.src.math.log10 libc.src.math.log10
libc.src.math.log10f libc.src.math.log10f
libc.src.math.log1pf libc.src.math.log1pf
libc.src.math.log2
libc.src.math.log2f libc.src.math.log2f
libc.src.math.log libc.src.math.log
libc.src.math.logf libc.src.math.logf
libc.src.math.logb libc.src.math.logb
libc.src.math.logbf libc.src.math.logbf
libc.src.math.logbl libc.src.math.logbl
libc.src.math.modf libc.src.math.modf
libc.src.math.modff libc.src.math.modff
Show All 39 Lines

libc/config/linux/aarch64/entrypoints.txt

Show First 20 Lines • Show All 282 Lines • ▼ Show 20 Linesset(TARGET_LIBM_ENTRYPOINTS
libc.src.math.lroundf libc.src.math.lroundf
libc.src.math.lroundl libc.src.math.lroundl
libc.src.math.ldexp libc.src.math.ldexp
libc.src.math.ldexpf libc.src.math.ldexpf
libc.src.math.ldexpl libc.src.math.ldexpl
libc.src.math.log10 libc.src.math.log10
libc.src.math.log10f libc.src.math.log10f
libc.src.math.log1pf libc.src.math.log1pf
libc.src.math.log2
libc.src.math.log2f libc.src.math.log2f
libc.src.math.log libc.src.math.log
libc.src.math.logf libc.src.math.logf
libc.src.math.logb libc.src.math.logb
libc.src.math.logbf libc.src.math.logbf
libc.src.math.logbl libc.src.math.logbl
libc.src.math.modf libc.src.math.modf
libc.src.math.modff libc.src.math.modff
▲ Show 20 Lines • Show All 188 Lines • Show Last 20 Lines

libc/config/linux/x86_64/entrypoints.txt

Show First 20 Lines • Show All 287 Lines • ▼ Show 20 Linesset(TARGET_LIBM_ENTRYPOINTS
libc.src.math.llrintf libc.src.math.llrintf
libc.src.math.llrintl libc.src.math.llrintl
libc.src.math.llround libc.src.math.llround
libc.src.math.llroundf libc.src.math.llroundf
libc.src.math.llroundl libc.src.math.llroundl
libc.src.math.log10 libc.src.math.log10
libc.src.math.log10f libc.src.math.log10f
libc.src.math.log1pf libc.src.math.log1pf
libc.src.math.log2
libc.src.math.log2f libc.src.math.log2f
libc.src.math.log libc.src.math.log
libc.src.math.logf libc.src.math.logf
libc.src.math.logb libc.src.math.logb
libc.src.math.logbf libc.src.math.logbf
libc.src.math.logbl libc.src.math.logbl
libc.src.math.lrint libc.src.math.lrint
libc.src.math.lrintf libc.src.math.lrintf
▲ Show 20 Lines • Show All 229 Lines • Show Last 20 Lines

libc/config/windows/entrypoints.txt

Show First 20 Lines • Show All 164 Lines • ▼ Show 20 Linesset(TARGET_LIBM_ENTRYPOINTS
libc.src.math.llrintf libc.src.math.llrintf
libc.src.math.llrintl libc.src.math.llrintl
libc.src.math.llround libc.src.math.llround
libc.src.math.llroundf libc.src.math.llroundf
libc.src.math.llroundl libc.src.math.llroundl
libc.src.math.log10 libc.src.math.log10
libc.src.math.log10f libc.src.math.log10f
libc.src.math.log1pf libc.src.math.log1pf
libc.src.math.log2
libc.src.math.log2f libc.src.math.log2f
libc.src.math.log libc.src.math.log
libc.src.math.logf libc.src.math.logf
libc.src.math.logb libc.src.math.logb
libc.src.math.logbf libc.src.math.logbf
libc.src.math.logbl libc.src.math.logbl
libc.src.math.lrint libc.src.math.lrint
libc.src.math.lrintf libc.src.math.lrintf
▲ Show 20 Lines • Show All 48 Lines • Show Last 20 Lines

libc/spec/stdc.td

Show First 20 Lines • Show All 404 Lines • ▼ Show 20 Lines HeaderSpec Math = HeaderSpec<
FunctionSpec<"ldexpf", RetValSpec<FloatType>, [ArgSpec<FloatType>, ArgSpec<IntType>]>, FunctionSpec<"ldexpf", RetValSpec<FloatType>, [ArgSpec<FloatType>, ArgSpec<IntType>]>,
FunctionSpec<"ldexpl", RetValSpec<LongDoubleType>, [ArgSpec<LongDoubleType>, ArgSpec<IntType>]>, FunctionSpec<"ldexpl", RetValSpec<LongDoubleType>, [ArgSpec<LongDoubleType>, ArgSpec<IntType>]>,
FunctionSpec<"log10", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>, FunctionSpec<"log10", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
FunctionSpec<"log10f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"log10f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"log1pf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"log1pf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"log2", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
FunctionSpec<"log2f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"log2f", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"log", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>, FunctionSpec<"log", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
FunctionSpec<"logf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"logf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"logb", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>, FunctionSpec<"logb", RetValSpec<DoubleType>, [ArgSpec<DoubleType>]>,
FunctionSpec<"logbf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>, FunctionSpec<"logbf", RetValSpec<FloatType>, [ArgSpec<FloatType>]>,
FunctionSpec<"logbl", RetValSpec<LongDoubleType>, [ArgSpec<LongDoubleType>]>, FunctionSpec<"logbl", RetValSpec<LongDoubleType>, [ArgSpec<LongDoubleType>]>,
▲ Show 20 Lines • Show All 695 Lines • Show Last 20 Lines

libc/src/math/CMakeLists.txt

Show First 20 Lines • Show All 110 Lines • ▼ Show 20 Lines
add_math_entrypoint_object(ldexpf) add_math_entrypoint_object(ldexpf)
add_math_entrypoint_object(ldexpl) add_math_entrypoint_object(ldexpl)
add_math_entrypoint_object(log10) add_math_entrypoint_object(log10)
add_math_entrypoint_object(log10f) add_math_entrypoint_object(log10f)
add_math_entrypoint_object(log1pf) add_math_entrypoint_object(log1pf)
add_math_entrypoint_object(log2)
add_math_entrypoint_object(log2f) add_math_entrypoint_object(log2f)
add_math_entrypoint_object(log) add_math_entrypoint_object(log)
add_math_entrypoint_object(logf) add_math_entrypoint_object(logf)
add_math_entrypoint_object(logb) add_math_entrypoint_object(logb)
add_math_entrypoint_object(logbf) add_math_entrypoint_object(logbf)
add_math_entrypoint_object(logbl) add_math_entrypoint_object(logbl)
▲ Show 20 Lines • Show All 66 Lines • Show Last 20 Lines

libc/src/math/generic/CMakeLists.txt

Show First 20 Lines • Show All 827 Lines • ▼ Show 20 Lines DEPENDS
libc.src.__support.FPUtil.fma libc.src.__support.FPUtil.fma
libc.src.__support.FPUtil.polyeval libc.src.__support.FPUtil.polyeval
libc.src.__support.macros.optimization libc.src.__support.macros.optimization
COMPILE_OPTIONS COMPILE_OPTIONS
-O3 -O3
) )
add_entrypoint_object( add_entrypoint_object(
log2
SRCS
log2.cpp
HDRS
../log2.h
DEPENDS
.common_constants
.log_range_reduction
libc.src.__support.FPUtil.fenv_impl
libc.src.__support.FPUtil.fp_bits
libc.src.__support.FPUtil.multiply_add
libc.src.__support.FPUtil.polyeval
libc.src.__support.FPUtil.double_double
libc.src.__support.FPUtil.dyadic_float
libc.src.__support.macros.optimization
COMPILE_OPTIONS
-O3
)
add_entrypoint_object(
log2f log2f
SRCS SRCS
log2f.cpp log2f.cpp
HDRS HDRS
../log2f.h ../log2f.h
DEPENDS DEPENDS
.common_constants .common_constants
libc.src.__support.FPUtil.except_value_utils libc.src.__support.FPUtil.except_value_utils
▲ Show 20 Lines • Show All 667 Lines • Show Last 20 Lines

libc/src/math/generic/log2.cpp

  • This file was added.
//===-- Double-precision log2(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/log2.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/dyadic_float.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/common.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "common_constants.h"
#include "log_range_reduction.h"
namespace __llvm_libc {
// 128-bit precision dyadic floating point numbers.
using Float128 = typename fputil::DyadicFloat<128>;
using MType = typename Float128::MantissaType;
namespace {
constexpr fputil::DoubleDouble LOG2_E = {0x1.777d0ffda0d24p-56,
0x1.71547652b82fep0};
// Extra errors from P is from using x^2 to reduce evaluation latency.
constexpr double P_ERR = 0x1.0p-49;
const fputil::DoubleDouble LOG_R1[128] = {
{0.0, 0.0},
{0x1.46662d417cedp-62, 0x1.010157588de71p-7},
{0x1.27c8e8416e71fp-60, 0x1.0205658935847p-6},
{-0x1.d192d0619fa67p-60, 0x1.8492528c8cabfp-6},
{0x1.c05cf1d753622p-59, 0x1.0415d89e74444p-5},
{-0x1.cdd6f7f4a137ep-59, 0x1.466aed42de3eap-5},
{0x1.a8be97660a23dp-60, 0x1.894aa149fb343p-5},
{-0x1.e48fb0500efd4p-59, 0x1.ccb73cdddb2ccp-5},
{-0x1.dd7009902bf32p-58, 0x1.08598b59e3a07p-4},
{-0x1.7558367a6acf6p-59, 0x1.1973bd1465567p-4},
{0x1.7a976d3b5b45fp-59, 0x1.3bdf5a7d1ee64p-4},
{0x1.f38745c5c450ap-58, 0x1.5e95a4d9791cbp-4},
{-0x1.72566212cdd05p-61, 0x1.700d30aeac0e1p-4},
{-0x1.478a85704ccb7p-58, 0x1.9335e5d594989p-4},
{-0x1.0057eed1ca59fp-59, 0x1.b6ac88dad5b1cp-4},
{0x1.a38cb559a6706p-58, 0x1.c885801bc4b23p-4},
{-0x1.a2bf991780d3fp-59, 0x1.ec739830a112p-4},
{-0x1.ac9f4215f9393p-58, 0x1.fe89139dbd566p-4},
{-0x1.0e63a5f01c691p-58, 0x1.1178e8227e47cp-3},
{-0x1.c6ef1d9b2ef7ep-59, 0x1.1aa2b7e23f72ap-3},
{-0x1.499a3f25af95fp-58, 0x1.2d1610c86813ap-3},
{0x1.7d411a5b944adp-58, 0x1.365fcb0159016p-3},
{-0x1.0d5604930f135p-58, 0x1.4913d8333b561p-3},
{-0x1.71a9682395bfdp-61, 0x1.527e5e4a1b58dp-3},
{-0x1.d34f0f4621bedp-60, 0x1.6574ebe8c133ap-3},
{-0x1.8de59c21e166cp-57, 0x1.6f0128b756abcp-3},
{-0x1.1232ce70be781p-57, 0x1.823c16551a3c2p-3},
{0x1.55aa8b6997a4p-58, 0x1.8beafeb38fe8cp-3},
{0x1.142c507fb7a3dp-58, 0x1.95a5adcf7017fp-3},
{0x1.bcafa9de97203p-57, 0x1.a93ed3c8ad9e3p-3},
{-0x1.6353ab386a94dp-57, 0x1.b31d8575bce3dp-3},
{0x1.dd355f6a516d7p-60, 0x1.bd087383bd8adp-3},
{0x1.60629242471a2p-57, 0x1.d1037f2655e7bp-3},
{0x1.aa11d49f96cb9p-58, 0x1.db13db0d4894p-3},
{0x1.2276041f43042p-59, 0x1.e530effe71012p-3},
{-0x1.08ab2ddc708ap-58, 0x1.ef5ade4dcffe6p-3},
{0x1.f665066f980a2p-57, 0x1.f991c6cb3b379p-3},
{0x1.cdb16ed4e9138p-56, 0x1.07138604d5862p-2},
{0x1.162c79d5d11eep-58, 0x1.0c42d676162e3p-2},
{-0x1.0e63a5f01c691p-57, 0x1.1178e8227e47cp-2},
{0x1.66fbd28b40935p-56, 0x1.16b5ccbacfb73p-2},
{-0x1.12aeb84249223p-57, 0x1.1bf99635a6b95p-2},
{0x1.e0efadd9db02bp-56, 0x1.269621134db92p-2},
{-0x1.82dad7fd86088p-56, 0x1.2bef07cdc9354p-2},
{-0x1.3d69909e5c3dcp-56, 0x1.314f1e1d35ce4p-2},
{-0x1.324f0e883858ep-58, 0x1.36b6776be1117p-2},
{-0x1.2ad27e50a8ec6p-56, 0x1.3c25277333184p-2},
{0x1.0dbb243827392p-57, 0x1.419b423d5e8c7p-2},
{0x1.8fb4c14c56eefp-60, 0x1.4718dc271c41bp-2},
{-0x1.123615b147a5dp-58, 0x1.4c9e09e172c3cp-2},
{-0x1.8f7e9b38a6979p-57, 0x1.522ae0738a3d8p-2},
{-0x1.0908d15f88b63p-57, 0x1.57bf753c8d1fbp-2},
{-0x1.6541148cbb8a2p-56, 0x1.5d5bddf595f3p-2},
{0x1.dc18ce51fff99p-57, 0x1.630030b3aac49p-2},
{0x1.a64eadd740178p-58, 0x1.68ac83e9c6a14p-2},
{0x1.657c222d868cdp-58, 0x1.6e60ee6af1972p-2},
{0x1.84a4ee3059583p-56, 0x1.741d876c67bb1p-2},
{-0x1.c168817443f22p-56, 0x1.79e26687cfb3ep-2},
{-0x1.219024acd3b77p-58, 0x1.7fafa3bd8151cp-2},
{-0x1.486666443b153p-56, 0x1.85855776dcbfbp-2},
{-0x1.70f2f38238303p-56, 0x1.8b639a88b2df5p-2},
{-0x1.ad4bb98c1f2c5p-56, 0x1.914a8635bf68ap-2},
{-0x1.89d2816cf838fp-57, 0x1.973a3431356aep-2},
{0x1.87bcbcfd3e187p-59, 0x1.9d32bea15ed3bp-2},
{-0x1.ba8062860ae23p-57, 0x1.a33440224fa79p-2},
{-0x1.ba8062860ae23p-57, 0x1.a33440224fa79p-2},
{0x1.bcafa9de97203p-56, 0x1.a93ed3c8ad9e3p-2},
{0x1.9d56c45dd3e86p-56, 0x1.af5295248cddp-2},
{0x1.494b610665378p-56, 0x1.b56fa04462909p-2},
{0x1.6fd02999b21e1p-59, 0x1.bb9611b80e2fbp-2},
{-0x1.bfc00b8f3feaap-56, 0x1.c1c60693fa39ep-2},
{-0x1.bfc00b8f3feaap-56, 0x1.c1c60693fa39ep-2},
{0x1.223eadb651b4ap-57, 0x1.c7ff9c74554c9p-2},
{0x1.0798270b29f39p-56, 0x1.ce42f18064743p-2},
{0x1.d7f4d3b3d406bp-56, 0x1.d490246defa6bp-2},
{-0x1.0b5837185a661p-56, 0x1.dae75484c9616p-2},
{-0x1.ac81cc8a4dfb8p-56, 0x1.e148a1a2726cep-2},
{-0x1.ac81cc8a4dfb8p-56, 0x1.e148a1a2726cep-2},
{0x1.57d646a17bc6ap-56, 0x1.e7b42c3ddad73p-2},
{-0x1.74b71fb5e57e3p-62, 0x1.ee2a156b413e5p-2},
{-0x1.0d487f5aba5e5p-57, 0x1.f4aa7ee03192dp-2},
{-0x1.0d487f5aba5e5p-57, 0x1.f4aa7ee03192dp-2},
{0x1.7e8f05924d259p-57, 0x1.fb358af7a4884p-2},
{0x1.1713a36138e19p-57, 0x1.00e5ae5b207abp-1},
{-0x1.17f9e54e78104p-57, 0x1.04360be7603adp-1},
{-0x1.17f9e54e78104p-57, 0x1.04360be7603adp-1},
{0x1.2241edf5fd1f7p-57, 0x1.078bf0533c568p-1},
{0x1.0d710fcfc4e0dp-55, 0x1.0ae76e2d054fap-1},
{0x1.0d710fcfc4e0dp-55, 0x1.0ae76e2d054fap-1},
{0x1.3300f002e836ep-55, 0x1.0e4898611cce1p-1},
{-0x1.91eee7772c7c2p-55, 0x1.11af823c75aa8p-1},
{-0x1.91eee7772c7c2p-55, 0x1.11af823c75aa8p-1},
{0x1.342eb628dba17p-56, 0x1.151c3f6f29612p-1},
{0x1.89df1568ca0bp-55, 0x1.188ee40f23ca6p-1},
{0x1.89df1568ca0bp-55, 0x1.188ee40f23ca6p-1},
{0x1.59bddae1ccce2p-56, 0x1.1c07849ae6007p-1},
{-0x1.2164ff40e9817p-56, 0x1.1f8635fc61659p-1},
{-0x1.2164ff40e9817p-56, 0x1.1f8635fc61659p-1},
{-0x1.fcc8dbccc25cbp-57, 0x1.230b0d8bebc98p-1},
{0x1.e0efadd9db02bp-55, 0x1.269621134db92p-1},
{0x1.e0efadd9db02bp-55, 0x1.269621134db92p-1},
{-0x1.6a0c343be95dcp-56, 0x1.2a2786d0ec107p-1},
{-0x1.b941ee770436bp-56, 0x1.2dbf557b0df43p-1},
{-0x1.b941ee770436bp-56, 0x1.2dbf557b0df43p-1},
{0x1.6c3a5f12642c9p-57, 0x1.315da4434068bp-1},
{0x1.6c3a5f12642c9p-57, 0x1.315da4434068bp-1},
{-0x1.f01ab6065515cp-56, 0x1.35028ad9d8c86p-1},
{0x1.21512aa596ea3p-55, 0x1.38ae2171976e7p-1},
{0x1.21512aa596ea3p-55, 0x1.38ae2171976e7p-1},
{0x1.1930603d87b6ep-56, 0x1.3c6080c36bfb5p-1},
{0x1.1930603d87b6ep-56, 0x1.3c6080c36bfb5p-1},
{0x1.86cf0f38b461ap-57, 0x1.4019c2125ca93p-1},
{-0x1.84f481051f71ap-56, 0x1.43d9ff2f923c5p-1},
{-0x1.84f481051f71ap-56, 0x1.43d9ff2f923c5p-1},
{0x1.2541aca7d5844p-55, 0x1.47a1527e8a2d3p-1},
{0x1.2541aca7d5844p-55, 0x1.47a1527e8a2d3p-1},
{0x1.c457b531506f6p-55, 0x1.4b6fd6f970c1fp-1},
{0x1.c457b531506f6p-55, 0x1.4b6fd6f970c1fp-1},
{0x1.d749362382a77p-56, 0x1.4f45a835a4e19p-1},
{0x1.d749362382a77p-56, 0x1.4f45a835a4e19p-1},
{0x1.988ba4aea614dp-56, 0x1.5322e26867857p-1},
{0x1.988ba4aea614dp-56, 0x1.5322e26867857p-1},
{0x1.80bff3303dd48p-55, 0x1.5707a26bb8c66p-1},
{0x1.80bff3303dd48p-55, 0x1.5707a26bb8c66p-1},
{-0x1.6714fbcd8135bp-55, 0x1.5af405c3649ep-1},
{-0x1.6714fbcd8135bp-55, 0x1.5af405c3649ep-1},
{0x1.1c066d235ee63p-56, 0x1.5ee82aa24192p-1},
{0.0, 0.0},
};
const LogRR LOG2_TABLE = {
// -log2(r) with 128-bit precision generated by SageMath with:
//
// for i in range(1, 127):
// r = 2^-8 * ceil( 2^8 * (1 - 2^(-8)) / (1 + i*2^(-7)) );
// s, m, e = RealField(128)(r).log2().sign_mantissa_exponent();
// print("{false,", e, ", MType({", hex(m % 2^64), ",", hex((m >> 64) %
// 2^64),
// "})},");
/* .step_1 = */ {
{false, 0, MType(0)},
{false, -134, MType({0xe8c251630adb856a, 0xb963dd107b993ada})},
{false, -133, MType({0xa41b08fbe05f82d0, 0xba1f7430f9aab1b2})},
{false, -132, MType({0x1f06c085bc1b865d, 0x8c25c7262b57c149})},
{false, -132, MType({0x2e1c07f0438ebac0, 0xbb9ca64ecac6aaef})},
{false, -132, MType({0xaacc0e21d6541224, 0xeb75e8f8ff5ff022})},
{false, -131, MType({0x31514aef39ce6303, 0x8dd9953002a4e866})},
{false, -131, MType({0x50799beaaab2940c, 0xa62b07f3457c4070})},
{false, -131, MType({0xda288fc615a727dc, 0xbeb024b67dda6339})},
{false, -131, MType({0x22dbbaced44516ce, 0xcb0657cd5dbe4f6f})},
{false, -131, MType({0xd939dceecdd9ce05, 0xe3da945b878e27d0})},
{false, -131, MType({0x9596a8e2e84c8f45, 0xfce4aee0e88b2749})},
{false, -130, MType({0x243efd9325954cfe, 0x84bf1c673032495d})},
{false, -130, MType({0x91d79938e7226384, 0x916d6e1559a4b696})},
{false, -130, MType({0x22563c9ed9462091, 0x9e37db2866f2850b})},
{false, -130, MType({0x3a53ca1181015ada, 0xa4a7c31dc6f9a5d5})},
{false, -130, MType({0x3eb8023eed65d601, 0xb19d45fa1be70855})},
{false, -130, MType({0xce5cabbd2d753d9b, 0xb823018e3cfc25f0})},
{false, -130, MType({0x54dbf16fb0695ee3, 0xc544c055fde99333})},
{false, -130, MType({0x5196a85a067c6739, 0xcbe0e589e3f6042d})},
{false, -130, MType({0xf349845e48955078, 0xd930124bea9a2c66})},
{false, -130, MType({0x815ef705cfaef035, 0xdfe33d3fffa66037})},
{false, -130, MType({0x2ba704dcaa76f41d, 0xed61169f220e97f2})},
{false, -130, MType({0x2062f36bc14d0d93, 0xf42be9e9b09b3def})},
{false, -129, MType({0x132880194144b02b, 0x80ecdde7d30ea2ed})},
{false, -129, MType({0x54880de63812fd49, 0x845e706cafd1bf61})},
{false, -129, MType({0xa87c02eaf36e2c29, 0x8b4e029b1f8ac391})},
{false, -129, MType({0x9804237ec8d9431d, 0x8ecc164ea93841ae})},
{false, -129, MType({0x20f81ca95d9e7968, 0x924e69589e6b6268})},
{false, -129, MType({0x124bc6f1acf95dc4, 0x995ff71b8773432d})},
{false, -129, MType({0x5a5e8e21bff3336b, 0x9cef470aacfb7bf9})},
{false, -129, MType({0x4e53fa3329f65894, 0xa08300be1f651473})},
{false, -129, MType({0x2742d7296a39eed6, 0xa7b7dd96762cc3c7})},
{false, -129, MType({0xf359c5544bc5e134, 0xab591735abc724e4})},
{false, -129, MType({0x6b6c874dd96e1d75, 0xaefee78f75707221})},
{false, -129, MType({0x21006678c0a5c390, 0xb2a95a4cc313bb59})},
{false, -129, MType({0x6d40900b25024b32, 0xb6587b432e47501b})},
{false, -129, MType({0x89e2eb553b279b3d, 0xbdc4f8167955698f})},
{false, -129, MType({0xd58525aad392ca50, 0xc1826c8608fe9951})},
{false, -129, MType({0x54dbf16fb0695ee3, 0xc544c055fde99333})},
{false, -129, MType({0x88d5eae3326327bb, 0xc90c004926e9dbfb})},
{false, -129, MType({0x46dfa05bddfded8c, 0xccd83954b6359379})},
{false, -129, MType({0xbfe9dbebf2e8a45e, 0xd47fcb8c0852f0c0})},
{false, -129, MType({0x7b11f1c5160c515c, 0xd85b3fa7a3407fa8})},
{false, -129, MType({0x1339e5677ec44dd0, 0xdc3be2bd8d837f7f})},
{false, -129, MType({0xea2b8c7bb0ee9c8b, 0xe021c2cf17ed9bdb})},
{false, -129, MType({0xaec562332791fe38, 0xe40cee16a2ff21c4})},
{false, -129, MType({0x71682ebacca79cfa, 0xe7fd7308d6895b14})},
{false, -129, MType({0xa5ad5ce9fb5a7bb6, 0xebf36055e1abc61e})},
{false, -129, MType({0x3225190531a852c5, 0xefeec4eac371584e})},
{false, -129, MType({0xda8ad649da21eab0, 0xf3efaff29c559a77})},
{false, -129, MType({0x4c3e2ea7c15c3d1e, 0xf7f630d808fc2ada})},
{false, -129, MType({0xbcb9bfa9852e0d35, 0xfc02574686680cc6})},
{false, -128, MType({0xce032f41d1e774e8, 0x800a1995f0019518})},
{false, -128, MType({0x9b39ffeebc29372a, 0x8215ea5cd3e4c4c7})},
{false, -128, MType({0x87f95f1befb6f806, 0x8424a6335c777e0b})},
{false, -128, MType({0xb987b42e3bb332a1, 0x8636557862acb7ce})},
{false, -128, MType({0x139a7ba83bf2d136, 0x884b00aef726cec5})},
{false, -128, MType({0x50799beaaab2941, 0x8a62b07f3457c407})},
{false, -128, MType({0x8bd744617e9b7d52, 0x8c7d6db7169e0cda})},
{false, -128, MType({0x46ad444333ceb10, 0x8e9b414b5a92a606})},
{false, -128, MType({0xef4c737fba4f5d66, 0x90bc345861bf3d52})},
{false, -128, MType({0xae441c09d761c549, 0x92e050231df57d6f})},
{false, -128, MType({0x6e36aa9ce90a3879, 0x95079e1a0382dc79})},
{false, -128, MType({0xefca1a184e93809, 0x973227d6027ebd8a})},
{false, -128, MType({0xefca1a184e93809, 0x973227d6027ebd8a})},
{false, -128, MType({0x124bc6f1acf95dc4, 0x995ff71b8773432d})},
{false, -128, MType({0x352bea51e58ea9e8, 0x9b9115db83a3dd2d})},
{false, -128, MType({0x266d6cdc959153bc, 0x9dc58e347d37696d})},
{false, -128, MType({0x4527d82c8214ddca, 0x9ffd6a73a78eaf35})},
{false, -128, MType({0x404cabb76d600e3c, 0xa238b5160413106e})},
{false, -128, MType({0x404cabb76d600e3c, 0xa238b5160413106e})},
{false, -128, MType({0xcab7d2ec23f0eef3, 0xa47778c98bcc86a1})},
{false, -128, MType({0x761c48dd859de2d3, 0xa6b9c06e6211646b})},
{false, -128, MType({0x7fd3b7d7e5d148bb, 0xa8ff971810a5e181})},
{false, -128, MType({0xc27c6780d92b4d11, 0xab49080ecda53208})},
{false, -128, MType({0xdb502402c94092cd, 0xad961ed0cb91d406})},
{false, -128, MType({0xdb502402c94092cd, 0xad961ed0cb91d406})},
{false, -128, MType({0x3432ef6b732b6843, 0xafe6e71393eeda29})},
{false, -128, MType({0xbb324da7e046e792, 0xb23b6cc56cc84c99})},
{false, -128, MType({0xb21709ce430c8e24, 0xb493bc0ec9954243})},
{false, -128, MType({0xb21709ce430c8e24, 0xb493bc0ec9954243})},
{false, -128, MType({0xe91ad16ecff10111, 0xb6efe153c7e319f6})},
{false, -128, MType({0xce31e481cd797e79, 0xb94fe935b83e3eb5})},
{false, -128, MType({0xda3e961a96c580fa, 0xbbb3e094b3d228d3})},
{false, -128, MType({0xda3e961a96c580fa, 0xbbb3e094b3d228d3})},
{false, -128, MType({0xf396598aae91499a, 0xbe1bd4913f3fda43})},
{false, -128, MType({0xae4cceb0f621941b, 0xc087d28dfb2febb8})},
{false, -128, MType({0xae4cceb0f621941b, 0xc087d28dfb2febb8})},
{false, -128, MType({0x6c1855c42078f81b, 0xc2f7e831632b6670})},
{false, -128, MType({0x169535fb8bf577c8, 0xc56c23679b4d206e})},
{false, -128, MType({0x169535fb8bf577c8, 0xc56c23679b4d206e})},
{false, -128, MType({0x3b24cecc60217942, 0xc7e492644d64237e})},
{false, -128, MType({0x3dc2687fcf939696, 0xca6143a49626d820})},
{false, -128, MType({0x3dc2687fcf939696, 0xca6143a49626d820})},
{false, -128, MType({0xa62e6add1a901a0, 0xcce245f1031e41fa})},
{false, -128, MType({0x5bb6e23138ad51e1, 0xcf67a85fa1f89a04})},
{false, -128, MType({0x5bb6e23138ad51e1, 0xcf67a85fa1f89a04})},
{false, -128, MType({0x7fc60a5103092bae, 0xd1f17a5621fb01ac})},
{false, -128, MType({0xbfe9dbebf2e8a45e, 0xd47fcb8c0852f0c0})},
{false, -128, MType({0xbfe9dbebf2e8a45e, 0xd47fcb8c0852f0c0})},
{false, -128, MType({0x8e2d7d378127d823, 0xd712ac0cf811659d})},
{false, -128, MType({0x5c1a7f14b168b365, 0xd9aa2c3b0ea3cbc1})},
{false, -128, MType({0x5c1a7f14b168b365, 0xd9aa2c3b0ea3cbc1})},
{false, -128, MType({0xb7579f0f8d3d514b, 0xdc465cd155a90942})},
{false, -128, MType({0xb7579f0f8d3d514b, 0xdc465cd155a90942})},
{false, -128, MType({0xb087205eb55aea85, 0xdee74ee64b0c38d3})},
{false, -128, MType({0x424a2623d60dfb16, 0xe18d13ee805a4de3})},
{false, -128, MType({0x424a2623d60dfb16, 0xe18d13ee805a4de3})},
{false, -128, MType({0x4d3a591ae6854787, 0xe437bdbf5254459c})},
{false, -128, MType({0x4d3a591ae6854787, 0xe437bdbf5254459c})},
{false, -128, MType({0x8dcdb6b24c5c5cdf, 0xe6e75e91b9cca551})},
{false, -128, MType({0x33ac7d9ebba8a53c, 0xe99c090536ece983})},
{false, -128, MType({0x33ac7d9ebba8a53c, 0xe99c090536ece983})},
{false, -128, MType({0xfb2eede4b59d8959, 0xec55d022d80e3d27})},
{false, -128, MType({0xfb2eede4b59d8959, 0xec55d022d80e3d27})},
{false, -128, MType({0x308b454666de8f99, 0xef14c7605d60654c})},
{false, -128, MType({0x308b454666de8f99, 0xef14c7605d60654c})},
{false, -128, MType({0x8383cb0ce23bebd4, 0xf1d902a37aaa5085})},
{false, -128, MType({0x8383cb0ce23bebd4, 0xf1d902a37aaa5085})},
{false, -128, MType({0x64fc87b4a41f7b70, 0xf4a2964538813c67})},
{false, -128, MType({0x64fc87b4a41f7b70, 0xf4a2964538813c67})},
{false, -128, MType({0x3f5d7d82b65c5686, 0xf77197157665f689})},
{false, -128, MType({0x3f5d7d82b65c5686, 0xf77197157665f689})},
{false, -128, MType({0x6476077b9fbd41ae, 0xfa461a5e8f4b759d})},
{false, -128, MType({0x6476077b9fbd41ae, 0xfa461a5e8f4b759d})},
{false, -128, MType({0xe3909ffd0d61778, 0xfd2035e9221ef5d0})},
{false, 0, MType(0)},
},
// -log2(r) for the second step, generated by SageMath with:
//
// for i in range(-2^6, 2^7 + 1):
// r = 2^-16 * round( 2^16 / (1 + i*2^(-14)) );
// s, m, e = RealField(128)(r).log2().sign_mantissa_exponent();
// print("{true," if s == 1 else "{false,", e, ",
// MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})},");
/* .step_2 = */
{
{true, -135, MType({0xb5cfed58337e848a, 0xb906155918954401})},
{true, -135, MType({0xffaf2ac1b1d20910, 0xb6264958a3c7fa2b})},
{true, -135, MType({0x52521a3950ea2ed8, 0xb34671e439aa448e})},
{true, -135, MType({0xf87e1abdee10fd95, 0xb0668efb7ef48ab7})},
{true, -135, MType({0xfbd43bbcc24c5e43, 0xad86a09e185af0e8})},
{true, -135, MType({0x2f4f5d48f9796742, 0xaaa6a6cbaa8d57ce})},
{true, -135, MType({0x3477fd67c1cab6b3, 0xa7c6a183da375c3d})},
{true, -135, MType({0x7b4d33eb381fe558, 0xa4e690c64c0056f0})},
{true, -135, MType({0x3ce25e48cb498dea, 0xa2067492a48b5c43})},
{true, -135, MType({0x70b0fcc9e4330983, 0x9f264ce888773bed})},
{true, -135, MType({0xbc9e4267d3189b22, 0x9c4619c79c5e80bf})},
{true, -135, MType({0x5fb3d896326615c4, 0x9965db2f84d7705f})},
{true, -135, MType({0x178b58311e96d323, 0x9685911fe6740b02})},
{true, -135, MType({0x6bf8b6cf73d847, 0x93a53b9865c20b2a})},
{true, -135, MType({0x7019f6e64a580a02, 0x90c4da98a74ae561})},
{true, -135, MType({0xcb5733cf0eb4191d, 0x8de46e204f93c7f6})},
{true, -135, MType({0x56148d4fc5e415b6, 0x8b03f62f031d9ab8})},
{true, -135, MType({0xfe5370f425872623, 0x882372c46664feaf})},
{true, -135, MType({0x21b72a1457ee70d6, 0x8542e3e01de24ddf})},
{true, -135, MType({0xabff4f89968bed0b, 0x81aa211f1e332fcf})},
{true, -136, MType({0x86410a676480a5a7, 0xfd92f0cf88d75f24})},
{true, -136, MType({0x44280889021970e4, 0xf7d1886b2a876289})},
{true, -136, MType({0x32eb139d9812090d, 0xf21009106a42bc14})},
{true, -136, MType({0xbef9dd41e8e42810, 0xec4e72be90cd2d2d})},
{true, -136, MType({0x689d08ca6c7c3eb1, 0xe68cc574e6e1e5d7})},
{true, -136, MType({0x1ef259a7f69821d, 0xe0cb0132b5338423})},
{true, -136, MType({0xe22cea71b7bb8467, 0xdb0925f7446c13a9})},
{true, -136, MType({0xe5bb27303f542fe, 0xd54733c1dd2d0d04})},
{true, -136, MType({0x57453c8d5dc64ce1, 0xcf852a91c80f553f})},
{true, -136, MType({0x6cc7add1fc09ef92, 0xc9c30a664da33d56})},
{true, -136, MType({0xe678d7280de1c07f, 0xc400d33eb67081a7})},
{true, -136, MType({0x419bbeb2239bdc39, 0xbe3e851a4af6496d})},
{true, -136, MType({0xd4676d1d81755809, 0xb87c1ff853ab2631})},
{true, -136, MType({0xb69dfef7ac2e2890, 0xb2b9a3d818fd1349})},
{true, -136, MType({0x9f72fa0a8fccabc0, 0xacf710b8e3517548})},
{true, -136, MType({0xb8bfe6a3addb988e, 0xa7346699fb051978})},
{true, -136, MType({0x67862c8ec9dcd60d, 0xa171a57aa86c3551})},
{true, -136, MType({0x9bd3370909e28a6, 0x9baecd5a33d265ee})},
{true, -136, MType({0xa96bc611b991419b, 0x95ebde37e57aaf84})},
{true, -136, MType({0xa50bb80f203f0d62, 0x9028d813059f7cdc})},
{true, -136, MType({0x4d36cd474f65a317, 0x8a65baeadc729ec5})},
{true, -136, MType({0x779be241ef4874a3, 0x84a286beb21d4b8c})},
{true, -137, MType({0xe76a962fa65ace3, 0xfdbe771b9d803cea})},
{true, -137, MType({0xd3d35627464a5267, 0xf237b2aef4e62e5a})},
{true, -137, MType({0x162ef4b0e838c363, 0xe6b0c035fa8b328c})},
{true, -137, MType({0x77bb10b976b3b9ca, 0xdb299faf3e7cd74f})},
{true, -137, MType({0x209853cee70bc58b, 0xcfa2511950b77014})},
{true, -137, MType({0x63f9b57cbaf2e58d, 0xc41ad472c12614d3})},
{true, -137, MType({0x4fca1c931bd6e6d6, 0xb89329ba1fa2a0fd})},
{true, -137, MType({0x26d26e434a53490a, 0xad0b50edfbf5b265})},
{true, -137, MType({0xc55e079078dc86a0, 0xa1834a0ce5d6a82d})},
{true, -137, MType({0xf05b9d5bd28f540b, 0x95fb15156ceba1b5})},
{true, -137, MType({0x8ef87f1a11cdb727, 0x8a72b20620c97d84})},
{true, -138, MType({0x9d6870114c1183cf, 0xfdd441bb21e7b069})},
{true, -138, MType({0x63d514fff97e86f3, 0xe6c2c33499ba16c4})},
{true, -138, MType({0x11a381901eadd883, 0xcfb0e875c7cc5929})},
{true, -138, MType({0xa9d69d37bc0a5bac, 0xb89eb17bcabe1857})},
{true, -138, MType({0x2dc97c9ffefd2497, 0xa18c1e43c10c6898})},
{true, -138, MType({0xdcdc8afcb2ac09a, 0x8a792ecac911cf92})},
{true, -139, MType({0xdd454eb3a1489470, 0xe6cbc61c020c8446})},
{true, -139, MType({0x878035864d84b319, 0xb8a476150dfe4470})},
{true, -139, MType({0x7ce595cc53b8342c, 0x8a7c6d7af1de7942})},
{true, -140, MType({0x4710b59049899141, 0xb8a7588fd29b1baa})},
{true, -141, MType({0x5957f633309d74e3, 0xb8a8c9d8be9ae994})},
{false, 0, MType({0x0, 0x0})},
{false, -141, MType({0x8268aba030b1adf6, 0xb8abac81ab576f3b})},
{false, -140, MType({0x1511cba2fb213a10, 0xb8ad1de1ac9ea6a5})},
{false, -139, MType({0x6379fb9fd9bc6235, 0x8a82eb7708262500})},
{false, -139, MType({0xb6fe1bf601ee27d5, 0xb8b000b8c65957cc})},
{false, -139, MType({0x8c6e60693a14e6d0, 0xe6ddcebbd72d3f7f})},
{false, -138, MType({0xe9bcfd0c62eaa2ca, 0x8a862ac30095c084})},
{false, -138, MType({0x73b214209a5234a7, 0xa19dca8e85918b6d})},
{false, -138, MType({0x347d4ca3109fe4db, 0xb8b5c6c35e142a9b})},
{false, -138, MType({0x37a62c48783bb066, 0xcfce1f646dca7745})},
{false, -138, MType({0x794b6437fb56344, 0xe6e6d4749883fbe3})},
{false, -138, MType({0x1cb9a45ed90318e6, 0xfdffe5f6c232f658})},
{false, -137, MType({0xbc118e5dbbef7dbc, 0x8a8ca9f6e7762d0f})},
{false, -137, MType({0xb4c0fb9535907cf8, 0x96198f2e5173e93b})},
{false, -137, MType({0xc051d2c5f00a9bb9, 0xa1a6a2a3113fe246})},
{false, -137, MType({0x553269878c1e5110, 0xad33e4569918a8d5})},
{false, -137, MType({0xbc906750b0ce372c, 0xb8c1544a5b4e2caf})},
{false, -137, MType({0x4c50eaa63be294b6, 0xc44ef27fca41bdd8})},
{false, -137, MType({0xb6cb28db8c065b44, 0xcfdcbef858660da1})},
{false, -137, MType({0x70479336830ceb05, 0xdb6ab9b5783f2fc5})},
{false, -137, MType({0x2a458c831f6aeb49, 0xe6f8e2b89c629b7a})},
{false, -137, MType({0x6489ba5bd391e206, 0xf2873a0337772c8a})},
{false, -137, MType({0x13f6fda510aeec3b, 0xfe15bf96bc35246b})},
{false, -136, MType({0x2f9a0ef9e8250836, 0x84d239ba4eb315a9})},
{false, -136, MType({0x389019e822b70f1e, 0x8a99aacf26f2a8a7})},
{false, -136, MType({0x308beeffa12cf669, 0x9061330aa04f87ae})},
{false, -136, MType({0x9886a71b25a2085d, 0x9628d26d7448a43f})},
{false, -136, MType({0x70ba9cebe0b969c3, 0x9bf088f85c65a56b})},
{false, -136, MType({0xcd855dc705ea2bea, 0xa1b856ac1236e85b})},
{false, -136, MType({0x7736196b11afb331, 0xa7803b894f5580e0})},
{false, -136, MType({0x94c99761b8eab3d8, 0xad483790cd6339fa})},
{false, -136, MType({0x6194b8c040814736, 0xb3104ac3460a9668})},
{false, -136, MType({0xedde8d24c7a999cc, 0xb8d8752172fed130})},
{false, -136, MType({0xea6b01ebde42f1d0, 0xbea0b6ac0dfbde2f})},
{false, -136, MType({0x7ef732b69334cf50, 0xc4690f63d0c66aa1})},
{false, -136, MType({0x2ba86275fcfc2d72, 0xca317f49752bddae})},
{false, -136, MType({0xb56ea44e185bf99f, 0xcffa065db50258f6})},
{false, -136, MType({0x1d5c3bbeb6902bfe, 0xd5c2a4a14a28b920})},
{false, -136, MType({0xa2f2bb9e156b0f37, 0xdb8b5a14ee86965f})},
{false, -136, MType({0xd166eb8da06ab5ef, 0xe15426b95c0c4506})},
{false, -136, MType({0x97dc7bae4219de0f, 0xe71d0a8f4cb2d60f})},
{false, -136, MType({0x6c9a8e7698f416c4, 0xece605977a7c17a8})},
{false, -136, MType({0x7b3a20aa5289695e, 0xf2af17d29f7295c0})},
{false, -136, MType({0xddcf578ee2c2897b, 0xf878414175a99a93})},
{false, -136, MType({0xe10ebd96c3ec30ec, 0xfe4181e4b73d2f37})},
{false, -135, MType({0xa9b7baecb34ba577, 0x82056cde8f290e13})},
{false, -135, MType({0x2da910dc61c182da, 0x8430f56d5e1edfd1})},
{false, -135, MType({0xfaca09dc7e0ba8b5, 0x8715b5a8f27bed90})},
{false, -135, MType({0xd723876173c0947, 0x89fa818019a2cace})},
{false, -135, MType({0x4e6651df154e8f8c, 0x8cdf58f330b64515})},
{false, -135, MType({0xee54b77d3bc34b6d, 0x8fc43c0294dd8af3})},
{false, -135, MType({0xad07dde9b5f92cce, 0x92a92aaea3442c3d})},
{false, -135, MType({0x261aacf944b638f0, 0x958e24f7b91a1a53})},
{false, -135, MType({0x232f5d64a85b219d, 0x98732ade3393a868})},
{false, -135, MType({0xf3a958bb706093fc, 0x9b583c626fe98bc9})},
{false, -135, MType({0xc9eaa059e7b0333a, 0x9e3d5984cb58dc25})},
{false, -135, MType({0x1e154029663243c0, 0xa1228245a32313cf})},
{false, -135, MType({0x16515200e283d006, 0xa407b6a5548e1006})},
{false, -135, MType({0xf498168a3337ca4f, 0xa6ecf6a43ce4113d})},
{false, -135, MType({0x8a04a89f0548a10f, 0xa9d24242b973bb63})},
{false, -135, MType({0xafaad01f25772805, 0xacb7998127901623})},
{false, -135, MType({0xc4f47950543fe0b8, 0xaf9cfc5fe4908d31})},
{false, -135, MType({0x338655e677d0d3ec, 0xb2826adf4dd0f08e})},
{false, -135, MType({0xf8ac2ce19d009541, 0xb567e4ffc0b174cc})},
{false, -135, MType({0x344d5e7dd7b2f465, 0xb84d6ac19a96b35c})},
{false, -135, MType({0xbd6a217fb4598ec7, 0xbb32fc2538e9aaca})},
{false, -135, MType({0xbc21ff368f562b75, 0xbe18992af917bf0e})},
{false, -135, MType({0x4944139ccbf2cb9a, 0xc0fe41d33892b9cc})},
{false, -135, MType({0x1369970c8b67e6b5, 0xc3e3f61e54d0ca9c})},
{false, -135, MType({0x99b370e2d04a530, 0xc6c9b60cab4c8752})},
{false, -135, MType({0xb81c3d48aff589f, 0xc9af819e9984ec44})},
{false, -135, MType({0x9f22b80993be311b, 0xcc9558d47cfd5c90})},
{false, -135, MType({0xac29209c8d8985ae, 0xcf7b3baeb33da265})},
{false, -135, MType({0x3cbb6a520292351d, 0xd2612a2d99d1ef47})},
{false, -135, MType({0x43de9ae40507ef24, 0xd54724518e4adc56})},
{false, -135, MType({0x69677b902ea4df3a, 0xd82d2a1aee3d6a97})},
{false, -135, MType({0xdb7a3aff74967bd5, 0xdb133b8a17430339})},
{false, -135, MType({0x25990c82a0066ac6, 0xddf9589f66f977de})},
{false, -135, MType({0xd424aacf4babf55, 0xe0df815b3b0302dd})},
{false, -135, MType({0xf8e3e7eb5a7bdebb, 0xe30c278d9936c595})},
{false, -135, MType({0x5ef8bf5adf5deebe, 0xe5f264adb62d5810})},
{false, -135, MType({0x331d19965368fc82, 0xe8d8ad75590bdf92})},
{false, -135, MType({0x901c30c427e358b8, 0xebbf01e4df85219e})},
{false, -135, MType({0xaeac7e9857253b06, 0xeea561fca7504dc1})},
{false, -135, MType({0xe2113e5893ab5b40, 0xf18bcdbd0e28fdd7})},
{false, -135, MType({0x9a4efc80ae977826, 0xf472452671cf3654})},
{false, -135, MType({0x6bf3ba8319332c9f, 0xf758c83930076689})},
{false, -135, MType({0x1d732d302e75018b, 0xfa3f56f5a69a68ed})},
{false, -135, MType({0xba179c5dbcceec01, 0xfd25f15c33558362})},
{false, -134, MType({0x5543f53b8ad85039, 0x80064bb69a0533c0})},
{false, -134, MType({0xe971a5565b93cb67, 0x8179a4948347996b})},
{false, -134, MType({0x5b399644ba714691, 0x82ed0348045f379d})},
{false, -134, MType({0x5079f1e0ec4b8496, 0x846067d14c3b8982})},
{false, -134, MType({0x6aba4990a32e8873, 0x85d3d23089ce40b0})},
{false, -134, MType({0xe16770c3a404291c, 0x87474265ec0b4548})},
{false, -134, MType({0x1edb7ffb1d6b3eab, 0x88bab871a1e8b61c})},
{false, -134, MType({0x603243e1ba7c7865, 0x8a2e3453da5ee8cd})},
{false, -134, MType({0x57ea5c03ea4621dd, 0x8ba1b60cc46869f6})},
{false, -134, MType({0xd3534cbf43bd7fd8, 0x8d153d9c8f01fd4a})},
{false, -134, MType({0x62c8c8075dc91cd5, 0x8e88cb03692a9dbc})},
{false, -134, MType({0x4bb70a5e3db7b85, 0x8ffc5e4181e37d9e})},
{false, -134, MType({0xd3875ba32159547a, 0x916ff757083006c7})},
{false, -134, MType({0x5c94c80e7a8f66b1, 0x9286adfca91ba28d})},
{false, -134, MType({0x52d313c47b4f91db, 0x93fa514ba0517623})},
{false, -134, MType({0x80829e9f3957a4c3, 0x956dfa72866fc57d})},
{false, -134, MType({0x1cd4917972015ae7, 0x96e1a9718a824be5})},
{false, -134, MType({0x1af23c29ef3032da, 0x98555e48db96fcd2})},
{false, -134, MType({0xe7f7bf240be67b80, 0x99c918f8a8be040e})},
{false, -134, MType({0x2bbe3cd4f7d868fa, 0x9b3cd9812109c5dc})},
{false, -134, MType({0x8c75d6a4c5ae460d, 0x9cb09fe2738edf14})},
{false, -134, MType({0x750fb989c9a06186, 0x9e246c1ccf642550})},
{false, -134, MType({0xde787e244901bdf9, 0x9f983e3063a2a709})},
{false, -134, MType({0x1ba3205ff729efa4, 0xa10c161d5f65abc0})},
{false, -134, MType({0xa864d2a038fb19cd, 0xa27ff3e3f1cab41b})},
{false, -134, MType({0xfb21f083a5fec56d, 0xa3f3d78449f17a11})},
{false, -134, MType({0x594c5552bcc377f5, 0xa567c0fe96fbf109})},
{false, -134, MType({0xaeb35a353fc5a503, 0xa6dbb053080e45fc})},
{false, -134, MType({0x67a5c05130c0f330, 0xa84fa581cc4edf9f})},
{false, -134, MType({0x4de5cafde1caf46f, 0xa9c3a08b12e65e81})},
{false, -134, MType({0x686fce3d160e88fd, 0xab37a16f0aff9d32})},
{false, -134, MType({0xde1375b3af6749a6, 0xacaba82de3c7b066})},
{false, -134, MType({0x243569048ac4affe, 0xadc2b114c632da56})},
{false, -134, MType({0xd6796227dcd39551, 0xaf36c21319b80ea2})},
{false, -134, MType({0xabc9265386172074, 0xb0aad8eccfb38d51})},
{false, -134, MType({0xcaac9f17896f2ce, 0xb21ef5a2175ac65e})},
{false, -134, MType({0x1c65a3c7f828972b, 0xb39318331fe56492})},
{false, -134, MType({0xabdc66446a4286d9, 0xb50740a0188d4daa})},
{false, -134, MType({0x2f3bbe8e8d72abec, 0xb67b6ee9308ea27b})},
{false, -134, MType({0xb67dbdd7f03d168c, 0xb7efa30e9727bf11})},
},
// -log2(r) for the third step, generated by SageMath with:
//
// for i in range(-80, 81):
// r = 2^-21 * round( 2^21 / (1 + i*2^(-21)) );
// s, m, e = RealField(128)(r).log2().sign_mantissa_exponent();
// print("{true," if (s == 1) else "{false,", e, ",
// MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})},");
/* .step_3 = */
{
{true, -142, MType({0x26f2c63c0827ccbb, 0xe6d3a96b978fc16e})},
{true, -142, MType({0x4b56fe667c8ec091, 0xe3f107a9fbfc50ca})},
{true, -142, MType({0x647d76181aec10fc, 0xe10e65d14b937265})},
{true, -142, MType({0x99e8f4d5379eca79, 0xde2bc3e18653b4f5})},
{true, -142, MType({0xf07da89990c20623, 0xdb4921daac3ba730})},
{true, -142, MType({0x4a8121848531851a, 0xd8667fbcbd49d7cd})},
{true, -142, MType({0x679a4d854ae13619, 0xd583dd87b97cd580})},
{true, -142, MType({0xe4d174072487a514, 0xd2a13b3ba0d32eff})},
{true, -142, MType({0x3c90319d969b54be, 0xcfbe98d8734b7301})},
{true, -142, MType({0xc6a173b09ba301e6, 0xccdbf65e30e43039})},
{true, -142, MType({0xb8317428d7d8d06b, 0xc9f953ccd99bf55e})},
{true, -142, MType({0x23cdb51bcc2061cd, 0xc716b1246d715125})},
{true, -142, MType({0xf964fc78084fd515, 0xc4340e64ec62d241})},
{true, -142, MType({0x6474fb15ccbb015, 0xc1516b8e566f076a})},
{true, -142, MType({0xf525ef6d0b75b1c3, 0xbe6ec8a0ab947f51})},
{true, -142, MType({0x4e13532df7ee8da7, 0xbb8c259bebd1c8ae})},
{true, -142, MType({0x76832500d72a9027, 0xb8a9828017257233})},
{true, -142, MType({0xb14a3d285e592ba0, 0xb5c6df4d2d8e0a95})},
{true, -142, MType({0x1e9e9dc9711f6e20, 0xb2e43c032f0a2089})},
{true, -142, MType({0xbc176e974f255fac, 0xb00198a21b9842c1})},
{true, -142, MType({0x64acf87fc0f648e6, 0xad1ef529f336fff3})},
{true, -142, MType({0xd0b8a1574433e1f8, 0xaa3c519ab5e4e6d1})},
{true, -142, MType({0x95f4e785371c69a9, 0xa759adf463a08610})},
{true, -142, MType({0x277d5db00363a46f, 0xa4770a36fc686c63})},
{true, -142, MType({0xd5cea669485ec36c, 0xa1946662803b287c})},
{true, -142, MType({0xcec66fda04833322, 0x9eb1c276ef174910})},
{true, -142, MType({0x1da36f6ebe3851db, 0x9bcf1e7448fb5cd2})},
{true, -142, MType({0xab055d83abfc0d82, 0x98ec7a5a8de5f273})},
{true, -142, MType({0x3cecf110dbda68e9, 0x9609d629bdd598a8})},
{true, -142, MType({0x76bbdb565a37e84b, 0x932731e1d8c8de22})},
{true, -142, MType({0xd934c38857eee4f3, 0x90448d82debe5194})},
{true, -142, MType({0xc27b427b4fbfc7db, 0x8d61e90ccfb481b1})},
{true, -142, MType({0x6e13de502b142b39, 0x8a7f447faba9fd2b})},
{true, -142, MType({0xf4e406206614e2ba, 0x879c9fdb729d52b3})},
{true, -142, MType({0x4d320daa3312ea6c, 0x84b9fb20248d10fd})},
{true, -142, MType({0x4aa528fc9d433c1a, 0x81d7564dc177c6b9})},
{true, -143, MType({0x3c8ad047559b1622, 0xfde962c892b80533})},
{true, -143, MType({0xacf765a8fc5bcc31, 0xf82418c77870a69f})},
{true, -143, MType({0xbe238832edd27f20, 0xf25ece9834168f1a})},
{true, -143, MType({0x2644bfca329b708, 0xec99843ac5a6dc07})},
{true, -143, MType({0xc6d05a788e614744, 0xe6d439af2d1eaac6})},
{true, -143, MType({0x133fe9cc57a8c1d0, 0xe10eeef56a7b18bc})},
{true, -143, MType({0xaa4cb429195fb5dd, 0xdb49a40d7db94348})},
{true, -143, MType({0x951ef239abbb959, 0xd58458f766d647ce})},
{true, -143, MType({0x686c430c89143d35, 0xcfbf0db325cf43ad})},
{true, -143, MType({0xba79c248afd42c12, 0xc9f9c240baa15447})},
{true, -143, MType({0xad19e0a92f115327, 0xc43476a0254996fd})},
{true, -143, MType({0xa8ad6ac3b0c99520, 0xbe6f2ad165c5292f})},
{true, -143, MType({0xd0567d4a9cc5e6a1, 0xb8a9ded47c11283d})},
{true, -143, MType({0x1f87c654b231443, 0xb2e492a9682ab188})},
{true, -143, MType({0xd6380b08358051bc, 0xad1f46502a0ee26d})},
{true, -143, MType({0xa07b024d26d391f6, 0xa759f9c8c1bad84e})},
{true, -143, MType({0x6ee868cb69e3a7d8, 0xa194ad132f2bb089})},
{true, -143, MType({0xa6869eff6682f73, 0x9bcf602f725e887d})},
{true, -143, MType({0xf6a44d559ccf3f61, 0x960a131d8b507d87})},
{true, -143, MType({0x72066e1d30a8e210, 0x9044c5dd79fead08})},
{true, -143, MType({0x75ba3245b1b856af, 0x8a7f786f3e66345c})},
{true, -143, MType({0xb5ac020473ab198f, 0x84ba2ad2d88430e1})},
{true, -144, MType({0x41127e3a88eb6741, 0xfde9ba1090ab7feb})},
{true, -144, MType({0xbf80787522aca1c4, 0xf25f1e1f1baffdea})},
{true, -144, MType({0xaf00688b14fa3adc, 0xe6d481d15210167b})},
{true, -144, MType({0x4d72837c8ab4d1e5, 0xdb49e52733c60457})},
{true, -144, MType({0x4e38ac27bb252090, 0xcfbf4820c0cc0236})},
{true, -144, MType({0xda3661f9292f59e8, 0xc434aabdf91c4ad0})},
{true, -144, MType({0x8fd0af9bdfd21488, 0xb8aa0cfedcb118de})},
{true, -144, MType({0x82ee19a9abf0bfa5, 0xad1f6ee36b84a716})},
{true, -144, MType({0x3cf68d5b5369a251, 0xa194d06ba591302f})},
{true, -144, MType({0xbcd34f38c977647e, 0x960a31978ad0eede})},
{true, -144, MType({0x76eee9c9605e2143, 0x8a7f92671b3e1dda})},
{true, -145, MType({0xaa6a3887f0c803ab, 0xfde9e5b4ada5efae})},
{true, -145, MType({0x6e25927e582ac191, 0xe6d4a5e27b136f13})},
{true, -145, MType({0xe2ebcac2f3a8e9eb, 0xcfbf65579eb92f4a})},
{true, -145, MType({0x9d9acc22d5690751, 0xb8aa2414188ba5bb})},
{true, -145, MType({0x1e12604b6d4132ef, 0xa194e217e87f47cb})},
{true, -145, MType({0xcf340d2acb9b92a9, 0x8a7f9f630e888add})},
{true, -146, MType({0xdc5e49fbde3c520, 0xe6d4b7eb1537c8ae})},
{true, -146, MType({0xc074c9557c01188, 0xb8aa2f9eb95b9332})},
{true, -146, MType({0xf0f82818ff9b654f, 0x8a7fa5e109656009})},
{true, -147, MType({0xd4cd612078bbe9b0, 0xb8aa35640a7c33eb})},
{true, -148, MType({0xf08cf68f42e09fa0, 0xb8aa3846b33aaecf})},
{false, 0, MType({0x0, 0x0})},
{false, -148, MType({0x68bd0facdf0ddaaf, 0xb8aa3e0c0513f9b1})},
{false, -147, MType({0x192af653dd41575b, 0xb8aa40eeae2ec9b3})},
{false, -146, MType({0x3b5c89842e540a51, 0x8a7fb2dd018e4892})},
{false, -146, MType({0x34ad8ebdd8b2750c, 0xb8aa46b400c0bee3})},
{false, -146, MType({0x70b12bd698e5be74, 0xe6d4dbfc54c5dd1b})},
{false, -145, MType({0x8c7e424efbd90e1, 0x8a7fb95afeda5c46})},
{false, -145, MType({0x31b8eba774a1de77, 0xa19505707dd23344})},
{false, -145, MType({0xee400e8c68838733, 0xb8aa523ea755fe32})},
{false, -145, MType({0xe71fa0b5603bc2f, 0xcfbf9fc57b7147be})},
{false, -145, MType({0x7763c919d8ac65f1, 0xe6d4ee04fa2f9a92})},
{false, -145, MType({0x232b270bb6046ec1, 0xfdea3cfd239c815e})},
{false, -144, MType({0x106f39197e068972, 0x8a7fc656fbe1c368})},
{false, -144, MType({0x4a4a6f4012941bd9, 0x960a6e8bbb581acc})},
{false, -144, MType({0x5bb34c1120b3e54b, 0xa195171cd0370c34})},
{false, -144, MType({0x6bb6731392a3147a, 0xad1fc00a3a845cf9})},
{false, -144, MType({0x2be1268dcee3c8fc, 0xb8aa6953fa45d275})},
{false, -144, MType({0xd84158d5d50251a9, 0xc43512fa0f813201})},
{false, -144, MType({0x3765bda15d0ef0fa, 0xcfbfbcfc7a3c40fa})},
{false, -144, MType({0x9a5ddb55f9cc27d9, 0xdb4a675b3a7cc4b9})},
{false, -144, MType({0xdcba1c593d918775, 0xe6d512165048829b})},
{false, -144, MType({0x648be060e1e30a95, 0xf25fbd2dbba53ffd})},
{false, -144, MType({0x22658dc2f1bcf6e8, 0xfdea68a17c98c23b})},
{false, -143, MType({0x48ad5162fb4a236e, 0x84ba8a38c9946759})},
{false, -143, MType({0xdb7fe3789405ce3a, 0x8a7fe04effad9560})},
{false, -143, MType({0x91b56e2e4f2e5ed8, 0x90453693609acde3})},
{false, -143, MType({0xf8998880c3bb4d76, 0x960a8d05ec5ef390})},
{false, -143, MType({0xe2b878052f67efee, 0x9bcfe3a6a2fce918})},
{false, -143, MType({0x67df399193f707c0, 0xa1953a758477912b})},
{false, -143, MType({0xe51b89e4d5d095e1, 0xa75a917290d1ce78})},
{false, -143, MType({0xfcbbee4edbf9f47d, 0xad1fe89dc80e83b1})},
{false, -143, MType({0x964fbd58b168371b, 0xb2e53ff72a309387})},
{false, -143, MType({0xdea7276ca7acd135, 0xb8aa977eb73ae0aa})},
{false, -143, MType({0x47d33f7e7afc83a6, 0xbe6fef346f304dcd})},
{false, -143, MType({0x892603b377909123, 0xc43547185213bda0})},
{false, -143, MType({0x9f32660aa06239fb, 0xc9fa9f2a5fe812d6})},
{false, -143, MType({0xcbcc5504d7407f6c, 0xcfbff76a98b03021})},
{false, -143, MType({0x9608c44d06402ebe, 0xd5854fd8fc6ef834})},
{false, -143, MType({0xca3db5604a863477, 0xdb4aa8758b274dc1})},
{false, -143, MType({0x7a024036206c37d6, 0xe110014044dc137c})},
{false, -143, MType({0xfc2e9be890ff7ee3, 0xe6d55a3929902c17})},
{false, -143, MType({0xecdc275c60da1b53, 0xec9ab36039467a47})},
{false, -143, MType({0x2d6571e94056607f, 0xf2600cb57401e0c0})},
{false, -143, MType({0xe4664401fd1ca2a7, 0xf8256638d9c54234})},
{false, -143, MType({0x7dbba7dcb50b3fd7, 0xfdeabfea6a93815a})},
{false, -142, MType({0xd541f90d853c794b, 0x81d80ce51337c072})},
{false, -142, MType({0xb08f65392ce8b75b, 0x84bab9ec06ae11c5})},
{false, -142, MType({0x6e969a29f8462436, 0x879d670a0fae2600})},
{false, -142, MType({0xcfc8cbcaa2bf130c, 0x8a80143f2e396e7d})},
{false, -142, MType({0xb737e48c19421e68, 0x8d62c18b62515c98})},
{false, -142, MType({0x2a9689b997c50c0b, 0x90456eeeabf761ac})},
{false, -142, MType({0x52381fccc774d66b, 0x93281c690b2cef13})},
{false, -142, MType({0x7910cec1dd92dc10, 0x960ac9fa7ff37629})},
{false, -142, MType({0xcb5866bbaff34cb, 0x98ed77a30a4c684a})},
{false, -142, MType({0x9d5c02c80c702d11, 0x9bd02562aa3936d0})},
{false, -142, MType({0xdddad0536b56e775, 0x9eb2d3395fbb5318})},
{false, -142, MType({0xa3a9505d7f71247a, 0xa19581272ad42e7e})},
{false, -142, MType({0xe6dfbd5d210830d7, 0xa4782f2c0b853a5d})},
{false, -142, MType({0xc2372f447bdcfa45, 0xa75add4801cfe812})},
{false, -142, MType({0x73099fd532c14b05, 0xaa3d8b7b0db5a8f9})},
{false, -142, MType({0x5951eef483de2c37, 0xad2039c52f37ee6e})},
{false, -142, MType({0xf7abe6ff6da76f1e, 0xb002e826665829cd})},
{false, -142, MType({0xf354411ed47c5d7b, 0xb2e5969eb317cc74})},
{false, -142, MType({0x1428a99ba8f5911f, 0xb5c8452e157847c0})},
{false, -142, MType({0x44a7c4330edff2c8, 0xb8aaf3d48d7b0d0c})},
{false, -142, MType({0x91f1306a84e4e07b, 0xbb8da2921b218db6})},
{false, -142, MType({0x2bc58de40cdf7b6a, 0xbe705166be6d3b1c})},
{false, -142, MType({0x648680b254df1d99, 0xc1530052775f869a})},
{false, -142, MType({0xb136b5ace0d6f74d, 0xc435af5545f9e18e})},
{false, -142, MType({0xa979e6c434fad480, 0xc7185e6f2a3dbd56})},
{false, -142, MType({0x794df5600c90a5a, 0xc9fb0da0242c8b50})},
{false, -142, MType({0xa86d80814ac18cf1, 0xccddbce833c7bcd8})},
{false, -142, MType({0x8b8ac57a9cca2d56, 0xcfc06c475910c34e})},
{false, -142, MType({0xd314c7e03140001f, 0xd2a31bbd9409100f})},
{false, -142, MType({0xc3d4c40e20b5ec89, 0xd585cb4ae4b2147a})},
{false, -142, MType({0xc5351d729060644e, 0xd8687aef4b0d41ed})},
{false, -142, MType({0x614162e1e12e445d, 0xdb4b2aaac71c09c7})},
{false, -142, MType({0x44a652eadf8ede85, 0xde2dda7d58dfdd66})},
{false, -142, MType({0x3eb1e02af3e52c3c, 0xe1108a67005a2e29})},
{false, -142, MType({0x415335a253a82aa2, 0xe3f33a67bd8c6d6f})},
{false, -142, MType({0x611abb0833305fe1, 0xe6d5ea7f90780c97})},
},
// -log2(r) for the fourth step, generated by SageMath with:
//
// for i in range(-65, 65):
// r = 2^-28 * round( 2^28 / (1 + i*2^(-28)) );
// s, m, e = RealField(128)(r).log2().sign_mantissa_exponent();
// print("{true," if (s == 1) else "{false,", e, ",
// MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), "})},");
/* .step_4 = */
{
{true, -149, MType({0xef1bffe565ce0a46, 0xbb8ce2990b5d0b90})},
{true, -149, MType({0xbea3244560ca3d99, 0xb8aa39b807a576e4})},
{true, -149, MType({0x8b91f71ceefa31a2, 0xb5c790d6d5c354df})},
{true, -149, MType({0x9096e3d684001c0e, 0xb2e4e7f575b6a57b})},
{true, -149, MType({0x86054c794367f36, 0xb0023f13e77f68b3})},
{true, -149, MType({0x2d9cb33094afe4de, 0xad1f96322b1d9e80})},
{true, -149, MType({0x3afa673cfb3698f3, 0xaa3ced50409146dd})},
{true, -149, MType({0x6b27d8033e4c6450, 0xa75a446e27da61c4})},
{true, -149, MType({0xf8d36b84d52a477b, 0xa4779b8be0f8ef2f})},
{true, -149, MType({0x1eab86ae37c03565, 0xa194f2a96becef1a})},
{true, -149, MType({0x175e8d56deb4ce2c, 0x9eb249c6c8b6617d})},
{true, -149, MType({0x1d9ae241436519da, 0x9bcfa0e3f7554653})},
{true, -149, MType({0x6c0ee71adfe44325, 0x98ecf800f7c99d96})},
{true, -149, MType({0x3d68fc7c2efb522f, 0x960a4f1dca136741})},
{true, -149, MType({0xcc5781e8ac28e749, 0x9327a63a6e32a34d})},
{true, -149, MType({0x5388d5ced3a0f5af, 0x9044fd56e42751b6})},
{true, -149, MType({0xdab5588224c7e4a, 0x8d6254732bf17275})},
{true, -149, MType({0x356d5d5915c94a70, 0x8a7fab8f45910584})},
{true, -149, MType({0x57d48712c69a6a7, 0x879d02ab31060ade})},
{true, -149, MType({0xb88970eae5341d60, 0x84ba59c6ee50827c})},
{true, -149, MType({0x89402fcbbfe331bb, 0x81d7b0e27d706c5a})},
{true, -150, MType({0x649fba0879ca348b, 0xfdea0ffbbccb90e3})},
{true, -150, MType({0xdccd9edfbab6f777, 0xf824be3222612d78})},
{true, -150, MType({0xf066b9aa4636478e, 0xf25f6c682ba1ae69})},
{true, -150, MType({0x14c7b3cb21578781, 0xec9a1a9dd88d13ab})},
{true, -150, MType({0xbf4d347b528f56e1, 0xe6d4c8d329235d30})},
{true, -150, MType({0x6553e0c9e1b70799, 0xe10f77081d648aef})},
{true, -150, MType({0x7c385b9bd80c1375, 0xdb4a253cb5509cdb})},
{true, -150, MType({0x795745ac402f919d, 0xd584d370f0e792e9})},
{true, -150, MType({0xd20d3d8c2625ac1b, 0xcfbf81a4d0296d0d})},
{true, -150, MType({0xfbb6dfa297551554, 0xc9fa2fd853162b3c})},
{true, -150, MType({0x6bb0c62ca2867d91, 0xc434de0b79adcd6b})},
{true, -150, MType({0x9757893d57e40877, 0xbe6f8c3e43f0538d})},
{true, -150, MType({0xf407bebdc8f8c28e, 0xb8aa3a70b1ddbd97})},
{true, -150, MType({0xf71dfa6d08b016be, 0xb2e4e8a2c3760b7e})},
{true, -150, MType({0x15f6cde02b5543ce, 0xad1f96d478b93d37})},
{true, -150, MType({0xc5eec8824692d1e9, 0xa75a4505d1a752b4})},
{true, -150, MType({0x7c6277947172081a, 0xa194f336ce404bec})},
{true, -150, MType({0xaeae662dc45a61ce, 0x9bcfa1676e8428d2})},
{true, -150, MType({0xd22f1d3b59110455, 0x960a4f97b272e95b})},
{true, -150, MType({0x5c4123804ab83462, 0x9044fdc79a0c8d7c})},
{true, -150, MType({0xc240fd95b5cecb89, 0x8a7fabf725511528})},
{true, -150, MType({0x798b2deab82fadc4, 0x84ba5a2654408055})},
{true, -151, MType({0xeef86988e2227ddb, 0xfdea10aa4db59ded})},
{true, -151, MType({0x62e1207c0209b090, 0xf25f6d073a400203})},
{true, -151, MType({0x3989789113ec7bee, 0xe6d4c9636e202cd4})},
{true, -151, MType({0x5daa65565e562909, 0xdb4a25bee9561e49})},
{true, -151, MType({0xb9fcd6062a84acbd, 0xcfbf8219abe1d64b})},
{true, -151, MType({0x3939b586c46792b3, 0xc434de73b5c354c4})},
{true, -151, MType({0xc619ea6a7a9ee85e, 0xb8aa3acd06fa999b})},
{true, -151, MType({0x4b5656ef9e7a27fd, 0xad1f97259f87a4bb})},
{true, -151, MType({0xb3a7d90083f7239c, 0xa194f37d7f6a760b})},
{true, -151, MType({0xe9c74a3381c0f016, 0x960a4fd4a6a30d75})},
{true, -151, MType({0xd86d7fcaf12ed012, 0x8a7fac2b15316ae2})},
{true, -152, MType({0xd4a6956a5c863e0f, 0xfdea1101962b1c76})},
{true, -152, MType({0x1462ef192f547877, 0xe6d4c9ab909eeed1})},
{true, -152, MType({0x45819d2f1d72eb8b, 0xcfbf825419be4ca6})},
{true, -152, MType({0x3d742790eedbe719, 0xb8aa3afb318935c8})},
{true, -152, MType({0xd1ac0d7b70d74492, 0xa194f3a0d7ffaa08})},
{true, -152, MType({0xd79ac58375f83d0c, 0x8a7fac450d21a939})},
{true, -153, MType({0x49637b2bac367e87, 0xe6d4c9cfa1de665a})},
{true, -153, MType({0x1cc4b5eedcc78b35, 0xb8aa3b1246d08f69})},
{true, -153, MType({0xd43bf48a42745836, 0x8a7fac520919cd43})},
{true, -154, MType({0x3557bdcf592619eb, 0xb8aa3b1dd1743f1c})},
{true, -155, MType({0x6bdc2e83d3ebb0c4, 0xb8aa3b2396c617ae})},
{false, 0, MType({0x0, 0x0})},
{false, -155, MType({0x2d5b40050e44e8ab, 0xb8aa3b2f2169ca44})},
{false, -154, MType({0xb8560371b8f04afe, 0xb8aa3b34e6bba447})},
{false, -153, MType({0xc79a43ccc70459cc, 0x8a7fac6c010a1f14})},
{false, -153, MType({0x22c25632f519f77f, 0xb8aa3b40715f59c0})},
{false, -153, MType({0x42c10a314e35fb9e, 0xe6d4ca17c45d8282})},
{false, -152, MType({0xbe5a212ed7b949e4, 0x8a7fac78fd024cdb})},
{false, -152, MType({0x12dcf94ef5c5b918, 0xa194f3e7892a4fde})},
{false, -152, MType({0x49781013e57110ce, 0xb8aa3b5786a6ca76})},
{false, -152, MType({0x8cba70c085c12cb3, 0xcfbf82c8f577bcd2})},
{false, -152, MType({0x7332f3fb09328b8, 0xe6d4ca3bd59d2721})},
{false, -152, MType({0xe37168243a9d8b14, 0xfdea11b02717098f})},
{false, -151, MType({0xa602205479b93722, 0x8a7fac92f4f2b226})},
{false, -151, MType({0xb5bd735852c0d583, 0x960a504e8f041bc3})},
{false, -151, MType({0x363248630b0d812d, 0xa194f40ae1bfc1b6})},
{false, -151, MType({0x3ca83f0e02b823c0, 0xad1f97c7ed25a415})},
{false, -151, MType({0xde66fb46974bc4fd, 0xb8aa3b85b135c2f7})},
{false, -151, MType({0x30b6254e23c69fc2, 0xc434df442df01e75})},
{false, -151, MType({0x48dd69ba009b370c, 0xcfbf83036354b6a4})},
{false, -151, MType({0x3c24797383b16af5, 0xdb4a26c351638b9c})},
{false, -151, MType({0x1fd309b800678db7, 0xe6d4ca83f81c9d74})},
{false, -151, MType({0x930d418c79378a3, 0xf25f6e45577fec43})},
{false, -151, MType({0xd85967b2783a12c, 0xfdea12076f8d7820})},
{false, -150, MType({0x210c898c360016ed, 0x84ba5ae52022a091})},
{false, -150, MType({0x5e19883eef2605ab, 0x8a7facc6e4d3a3b0})},
{false, -150, MType({0x488dacc6629300ae, 0x9044fea905d9c579})},
{false, -150, MType({0x6b0cdebd3264e3e3, 0x960a508b833505f7})},
{false, -150, MType({0x503b07e7ff788dc2, 0x9bcfa26e5ce56536})},
{false, -150, MType({0x82bc1435696a69d1, 0xa194f45192eae341})},
{false, -150, MType({0x8d33f1be0e96fb1f, 0xa75a463525458024})},
{false, -150, MType({0xfa4690c48c1b66c9, 0xad1f981913f53bea})},
{false, -150, MType({0x5497e3b57dd5fe75, 0xb2e4e9fd5efa16a0})},
{false, -150, MType({0x26cbdf277e66cad5, 0xb8aa3be206541050})},
{false, -150, MType({0xfb8679db27301625, 0xbe6f8dc70a032905})},
{false, -150, MType({0x5d6bacbb1056f6aa, 0xc434dfac6a0760cd})},
{false, -150, MType({0xd71f72dbd0c3d936, 0xc9fa31922660b7b1})},
{false, -150, MType({0xf345c97bfe230ba2, 0xcfbf83783f0f2dbe})},
{false, -150, MType({0x3c82b0042ce54751, 0xd584d55eb412c300})},
{false, -150, MType({0x3d7a2806f0403bae, 0xdb4a2745856b7781})},
{false, -150, MType({0x80d03540da2f18ae, 0xe10f792cb3194b4d})},
{false, -150, MType({0x9128dd987b73194f, 0xe6d4cb143d1c3e70})},
{false, -150, MType({0xf928291e63940e14, 0xec9a1cfc237450f5})},
{false, -150, MType({0x4372220d20e0e78a, 0xf25f6ee4662182e9})},
{false, -150, MType({0xfaaad4c9407040c7, 0xf824c0cd0523d455})},
{false, -150, MType({0xa9764fe14e20e9e4, 0xfdea12b6007b4547})},
{false, -149, MType({0xed3c5206ea4d3942, 0x81d7b24fac13eae4})},
{false, -149, MType({0xc2af218aea6da27, 0x84ba5b448614c2f4})},
{false, -149, MType({0xf6d912ac383aaeba, 0x879d04398e402ad6})},
{false, -149, MType({0x7298bf5cca8b3d95, 0x8a7fad2ec4962293})},
{false, -149, MType({0x44bc04daa8808214, 0x8d6256242916aa2f})},
{false, -149, MType({0x3294f0eb14683198, 0x9044ff19bbc1c1b0})},
{false, -149, MType({0x17592684ff600c3, 0x9327a80f7c97691c})},
{false, -149, MType({0x76aff9419c43e8b9, 0x960a51056b97a078})},
{false, -149, MType({0x5796367b39d26c63, 0x98ecf9fb88c267cb})},
{false, -149, MType({0x697a5c2e6888ddaa, 0x9bcfa2f1d417bf1a})},
{false, -149, MType({0x71ae7d8967b5a2b7, 0x9eb24be84d97a66b})},
{false, -149, MType({0x3584aecf760e7b39, 0xa194f4def5421dc4})},
{false, -149, MType({0x7a4f0558d1b0c59e, 0xa4779dd5cb17252a})},
{false, -149, MType({0x55f9792b821c455, 0xa75a46cccf16bca4})},
{false, -149, MType({0x9c087cff664ee311, 0xaa3cefc40140e436})},
{false, -149, MType({0x39bce36188dfc04, 0xad1f98bb61959be8})},
{false, -149, MType({0x16ba4e30a9d9d21, 0xb00241b2f014e3be})},
{false, -149, MType({0x5aca1bc777a54d5e, 0xb2e4eaaaacbebbbe})},
{false, -149, MType({0xd5094eb99a35d1f0, 0xb5c793a2979323ee})},
{false, -149, MType({0x357b5aa4ac49738d, 0xb8aa3c9ab0921c55})},
}};
// > P = fpminimax(log2(1 + x)/x, 3, [|128...|], [-0x1.0002143p-29 , 0x1p-29]);
// > P;
// > dirtyinfnorm(log2(1 + x)/x - P, [-0x1.0002143p-29 , 0x1p-29]);
// 0x1.27ad5...p-121
const Float128 BIG_COEFFS[4]{
{true, -129, MType({0x3eccf6940d66bbcc, 0xb8aa3b295c2b21e3})},
{false, -129, MType({0xee39a6d649394bb1, 0xf6384ee1d01febc9})},
{true, -128, MType({0xbe87fed067ea2ad5, 0xb8aa3b295c17f0bb})},
{false, -127, MType({0xbe87fed0691d3e3f, 0xb8aa3b295c17f0bb})},
};
// Reuse the output of the fast pass range reduction.
// -2^-8 <= m_x < 2^-7
double log2_accurate(int e_x, int index, double m_x) {
Float128 sum(static_cast<float>(e_x));
sum = fputil::quick_add(sum, LOG2_TABLE.step_1[index]);
Float128 v_f128 = log_range_reduction(m_x, LOG2_TABLE, sum);
// Polynomial approximation
Float128 p = fputil::quick_mul(v_f128, BIG_COEFFS[0]);
p = fputil::quick_mul(v_f128, fputil::quick_add(p, BIG_COEFFS[1]));
p = fputil::quick_mul(v_f128, fputil::quick_add(p, BIG_COEFFS[2]));
p = fputil::quick_mul(v_f128, fputil::quick_add(p, BIG_COEFFS[3]));
Float128 r = fputil::quick_add(sum, p);
return static_cast<double>(r);
}
} // namespace
LLVM_LIBC_FUNCTION(double, log2, (double x)) {
using FPBits_t = typename fputil::FPBits<double>;
FPBits_t xbits(x);
uint64_t x_u = xbits.uintval();
int x_e = -FPBits_t::EXPONENT_BIAS;
if (LIBC_UNLIKELY(x_u == 0x3FF0'0000'0000'0000ULL)) {
// log2(1.0) = +0.0
return 0.0;
}
if (LIBC_UNLIKELY(xbits.uintval() < FPBits_t::MIN_NORMAL ||
xbits.uintval() > FPBits_t::MAX_NORMAL)) {
if (xbits.is_zero()) {
// return -Inf and raise FE_DIVBYZERO.
fputil::set_errno_if_required(ERANGE);
fputil::raise_except_if_required(FE_DIVBYZERO);
return static_cast<double>(FPBits_t::neg_inf());
}
if (xbits.get_sign() && !xbits.is_nan()) {
fputil::set_errno_if_required(EDOM);
fputil::raise_except_if_required(FE_INVALID);
return FPBits_t::build_quiet_nan(0);
}
if (xbits.is_inf_or_nan()) {
return x;
}
// Normalize denormal inputs.
xbits.set_val(x * 0x1.0p52);
x_e -= 52;
x_u = xbits.uintval();
}
// log2(x) = log2(2^x_e * x_m)
// = x_e + log2(x_m)
// Range reduction for log2(x_m):
// For each x_m, we would like to find r such that:
// -2^-8 <= r * x_m - 1 < 2^-7
int shifted = x_u >> 45;
int index = shifted & 0x7F;
double r = RD[index];
// Add unbiased exponent. Add an extra 1 if the 8 leading fractional bits are
// all 1's.
x_e += static_cast<int>((x_u + (1ULL << 45)) >> 52);
double e_x = static_cast<double>(x_e);
// Set m = 1.mantissa.
uint64_t x_m = (x_u & 0x000F'FFFF'FFFF'FFFFULL) | 0x3FF0'0000'0000'0000ULL;
double m = FPBits_t(x_m).get_val();
double u, u_sq, err;
fputil::DoubleDouble r1;
// Perform exact range reduction
#ifdef LIBC_TARGET_CPU_HAS_FMA
u = fputil::multiply_add(r, m, -1.0); // exact
#else
uint64_t c_m = x_m & 0x3FFF'E000'0000'0000ULL;
double c = FPBits_t(c_m).get_val();
u = fputil::multiply_add(r, m - c, CD[index]); // exact
#endif // LIBC_TARGET_CPU_HAS_FMA
// Exact sum:
// r1.hi + r1.lo = e_x * log(2)_hi - log(r)_hi + u
r1 = fputil::exact_add(LOG_R1[index].hi, u);
// Error of u_sq = ulp(u^2);
u_sq = u * u;
// Total error is bounded by ~ C * ulp(u^2).
err = u_sq * P_ERR;
// Degree-7 minimax polynomial
double p0 = fputil::multiply_add(u, LOG_COEFFS[1], LOG_COEFFS[0]);
double p1 = fputil::multiply_add(u, LOG_COEFFS[3], LOG_COEFFS[2]);
double p2 = fputil::multiply_add(u, LOG_COEFFS[5], LOG_COEFFS[4]);
double p = fputil::polyeval(u_sq, LOG_R1[index].lo, p0, p1, p2);
r1.lo += p;
// Quick double-double multiplication:
// r2.hi + r2.lo ~ r1 * log2(e),
// with error bounded by:
// 4*ulp( ulp(r2.hi) )
fputil::DoubleDouble r2 = fputil::quick_mult(r1, LOG2_E);
fputil::DoubleDouble r3 = fputil::exact_add(e_x, r2.hi);
r3.lo += r2.lo;
// Overall, if we choose sufficiently large constant C, the total error is
// bounded by (C * ulp(u^2)).
// Lower bound from the result
double left = r3.hi + (r3.lo - err);
// Upper bound from the result
double right = r3.hi + (r3.lo + err);
// Ziv's test if fast pass is accurate enough.
if (left == right)
return left;
return log2_accurate(x_e, index, u);
}
} // namespace __llvm_libc

libc/src/math/log2.h

  • This file was added.
//===-- Implementation header for log2 --------------------------*- 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_LOG2_H
#define LLVM_LIBC_SRC_MATH_LOG2_H
namespace __llvm_libc {
double log2(double x);
} // namespace __llvm_libc
#endif // LLVM_LIBC_SRC_MATH_LOG2_H

libc/src/string/memory_utils/utils.h

Show All 33 Lines
} }
// Return whether `value` is a power of two. // Return whether `value` is a power of two.
static constexpr bool is_power2(size_t value) { static constexpr bool is_power2(size_t value) {
return value && is_power2_or_zero(value); return value && is_power2_or_zero(value);
} }
// Compile time version of log2 that handles 0. // Compile time version of log2 that handles 0.
static constexpr size_t log2(size_t value) { static constexpr size_t log2s(size_t value) {
return (value == 0 || value == 1) ? 0 : 1 + log2(value / 2); return (value == 0 || value == 1) ? 0 : 1 + log2s(value / 2);
} }
// Returns the first power of two preceding value or value if it is already a // Returns the first power of two preceding value or value if it is already a
// power of two (or 0 when value is 0). // power of two (or 0 when value is 0).
static constexpr size_t le_power2(size_t value) { static constexpr size_t le_power2(size_t value) {
return value == 0 ? value : 1ULL << log2(value); return value == 0 ? value : 1ULL << log2s(value);
} }
// Returns the first power of two following value or value if it is already a // Returns the first power of two following value or value if it is already a
// power of two (or 0 when value is 0). // power of two (or 0 when value is 0).
static constexpr size_t ge_power2(size_t value) { static constexpr size_t ge_power2(size_t value) {
return is_power2_or_zero(value) ? value : 1ULL << (log2(value) + 1); return is_power2_or_zero(value) ? value : 1ULL << (log2s(value) + 1);
} }
// Returns the number of bytes to substract from ptr to get to the previous // Returns the number of bytes to substract from ptr to get to the previous
// multiple of alignment. If ptr is already aligned returns 0. // multiple of alignment. If ptr is already aligned returns 0.
template <size_t alignment> uintptr_t distance_to_align_down(const void *ptr) { template <size_t alignment> uintptr_t distance_to_align_down(const void *ptr) {
static_assert(is_power2(alignment), "alignment must be a power of 2"); static_assert(is_power2(alignment), "alignment must be a power of 2");
return reinterpret_cast<uintptr_t>(ptr) & (alignment - 1U); return reinterpret_cast<uintptr_t>(ptr) & (alignment - 1U);
} }
▲ Show 20 Lines • Show All 222 Lines • Show Last 20 Lines

libc/test/src/math/CMakeLists.txt

Show First 20 Lines • Show All 1,290 Lines • ▼ Show 20 Linesadd_fp_unittest(
DEPENDS DEPENDS
libc.src.errno.errno libc.src.errno.errno
libc.include.math libc.include.math
libc.src.math.logf libc.src.math.logf
libc.src.__support.FPUtil.fp_bits libc.src.__support.FPUtil.fp_bits
) )
add_fp_unittest( add_fp_unittest(
log2_test
NEED_MPFR
SUITE
libc_math_unittests
SRCS
log2_test.cpp
DEPENDS
libc.src.errno.errno
libc.include.math
libc.src.math.log2
libc.src.__support.FPUtil.fp_bits
)
add_fp_unittest(
log2f_test log2f_test
NEED_MPFR NEED_MPFR
SUITE SUITE
libc_math_unittests libc_math_unittests
SRCS SRCS
log2f_test.cpp log2f_test.cpp
DEPENDS DEPENDS
libc.src.errno.errno libc.src.errno.errno
▲ Show 20 Lines • Show All 284 Lines • Show Last 20 Lines

libc/test/src/math/log2_test.cpp

  • This file was added.
//===-- Unittests for log2 ------------------------------------------------===//
//
// 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/errno/libc_errno.h"
#include "src/math/log2.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(double)
TEST(LlvmLibcLog2Test, SpecialNumbers) {
EXPECT_FP_EQ(aNaN, __llvm_libc::log2(aNaN));
EXPECT_FP_EQ(inf, __llvm_libc::log2(inf));
EXPECT_FP_IS_NAN_WITH_EXCEPTION(__llvm_libc::log2(neg_inf), FE_INVALID);
EXPECT_FP_EQ_WITH_EXCEPTION(neg_inf, __llvm_libc::log2(0.0), FE_DIVBYZERO);
EXPECT_FP_EQ_WITH_EXCEPTION(neg_inf, __llvm_libc::log2(-0.0), FE_DIVBYZERO);
EXPECT_FP_IS_NAN_WITH_EXCEPTION(__llvm_libc::log2(-1.0), FE_INVALID);
EXPECT_FP_EQ_ALL_ROUNDING(zero, __llvm_libc::log2(1.0));
}
TEST(LlvmLibcLog2Test, TrickyInputs) {
constexpr int N = 30;
constexpr uint64_t INPUTS[N] = {
0x3ff0000000000000, // x = 1.0
0x4024000000000000, // x = 10.0
0x4059000000000000, // x = 10^2
0x408f400000000000, // x = 10^3
0x40c3880000000000, // x = 10^4
0x40f86a0000000000, // x = 10^5
0x412e848000000000, // x = 10^6
0x416312d000000000, // x = 10^7
0x4197d78400000000, // x = 10^8
0x41cdcd6500000000, // x = 10^9
0x4202a05f20000000, // x = 10^10
0x42374876e8000000, // x = 10^11
0x426d1a94a2000000, // x = 10^12
0x42a2309ce5400000, // x = 10^13
0x42d6bcc41e900000, // x = 10^14
0x430c6bf526340000, // x = 10^15
0x4341c37937e08000, // x = 10^16
0x4376345785d8a000, // x = 10^17
0x43abc16d674ec800, // x = 10^18
0x43e158e460913d00, // x = 10^19
0x4415af1d78b58c40, // x = 10^20
0x444b1ae4d6e2ef50, // x = 10^21
0x4480f0cf064dd592, // x = 10^22
0x3fefffffffef06ad, 0x3fefde0f22c7d0eb, 0x225e7812faadb32f,
0x3fee1076964c2903, 0x3fdfe93fff7fceb0, 0x3ff012631ad8df10,
0x3fefbfdaa448ed98,
};
for (int i = 0; i < N; ++i) {
double x = double(FPBits(INPUTS[i]));
EXPECT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Log2, x,
__llvm_libc::log2(x), 0.5);
}
}
TEST(LlvmLibcLog2Test, AllExponents) {
double x = 0x1.0p-1074;
for (int i = -1074; i < 1024; ++i, x *= 2.0) {
ASSERT_MPFR_MATCH_ALL_ROUNDING(mpfr::Operation::Log2, x,
__llvm_libc::log2(x), 0.5);
}
}
TEST(LlvmLibcLog2Test, InDoubleRange) {
constexpr uint64_t COUNT = 1234561;
constexpr uint64_t START = 0x3FD0'0000'0000'0000ULL; // 0.25
constexpr uint64_t STOP = 0x4010'0000'0000'0000ULL; // 4.0
// constexpr uint64_t START = 0x3FF0'0000'0000'0000ULL; // 1.0
// constexpr uint64_t STOP = 0x4000'0000'0000'0000ULL; // 2.0
constexpr uint64_t STEP = (STOP - START) / COUNT;
auto test = [&](mpfr::RoundingMode rounding_mode) {
mpfr::ForceRoundingMode __r(rounding_mode);
uint64_t fails = 0;
uint64_t count = 0;
uint64_t cc = 0;
double mx, mr = 0.0;
double tol = 0.5;
for (uint64_t i = 0, v = START; i <= COUNT; ++i, v += STEP) {
double x = FPBits(v).get_val();
if (isnan(x) || isinf(x) || x < 0.0)
continue;
libc_errno = 0;
double result = __llvm_libc::log2(x);
++cc;
if (isnan(result) || isinf(result))
continue;
++count;
// ASSERT_MPFR_MATCH(mpfr::Operation::Log2, x, result, 0.5);
if (!EXPECT_MPFR_MATCH_ROUNDING_SILENTLY(mpfr::Operation::Log2, x, result,
0.5, rounding_mode)) {
++fails;
while (!EXPECT_MPFR_MATCH_ROUNDING_SILENTLY(
mpfr::Operation::Log2, x, result, tol, rounding_mode)) {
mx = x;
mr = result;
tol *= 2.0;
}
}
}
tlog << " Log2 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::Log2, 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);
}