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log10.cpp
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log10.cpp
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//===-- Double-precision log10(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/log10.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.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
namespace __llvm_libc {
// 192-bit precision dyadic floating point numbers.
using Float192 = typename fputil::DyadicFloat<192>;
using MType = typename Float192::MantissaType;
namespace {
// log10(2) generated by Sollya with:
// > a = round(log10(2), 43, RN);
constexpr double LOG10_2_HI = 0x1.34413509f78p-2; // LSB = 2^-43
// > b = round(log10(2) - a, D, RN);
constexpr double LOG10_2_LO = 0x1.fef311f12b358p-46; // LSB = 2^-98
constexpr double LOG10_2_ULP[2] = {0x1.0p-97, 0.0};
// Generated by Sollya with:
// > for i from 0 to 127 do {
// r = 2^-8 * nearestint( 2^8 * (1 + (i + 0.5)*2^(-7) - 2^(-15)) /
// ((1 + i*2^-7)*(1 + (i + 1)*2^-7)) );
// print(r, ",");
// };
// To improve the accuracy with inputs close to 1, we replace R[0] with 1.0.
constexpr double R[128] = {
1.0, 0x1.fap-1, 0x1.f6p-1, 0x1.f2p-1, 0x1.eep-1, 0x1.eap-1, 0x1.e8p-1,
0x1.e4p-1, 0x1.ep-1, 0x1.dcp-1, 0x1.dap-1, 0x1.d6p-1, 0x1.d2p-1, 0x1.dp-1,
0x1.ccp-1, 0x1.c8p-1, 0x1.c6p-1, 0x1.c2p-1, 0x1.cp-1, 0x1.bcp-1, 0x1.bap-1,
0x1.b6p-1, 0x1.b4p-1, 0x1.bp-1, 0x1.aep-1, 0x1.aap-1, 0x1.a8p-1, 0x1.a6p-1,
0x1.a2p-1, 0x1.ap-1, 0x1.9ep-1, 0x1.9ap-1, 0x1.98p-1, 0x1.96p-1, 0x1.94p-1,
0x1.9p-1, 0x1.8ep-1, 0x1.8cp-1, 0x1.8ap-1, 0x1.88p-1, 0x1.84p-1, 0x1.82p-1,
0x1.8p-1, 0x1.7ep-1, 0x1.7cp-1, 0x1.7ap-1, 0x1.78p-1, 0x1.76p-1, 0x1.74p-1,
0x1.72p-1, 0x1.7p-1, 0x1.6ep-1, 0x1.6cp-1, 0x1.6ap-1, 0x1.68p-1, 0x1.66p-1,
0x1.64p-1, 0x1.62p-1, 0x1.6p-1, 0x1.5ep-1, 0x1.5cp-1, 0x1.5ap-1, 0x1.58p-1,
0x1.56p-1, 0x1.54p-1, 0x1.52p-1, 0x1.5p-1, 0x1.5p-1, 0x1.4ep-1, 0x1.4cp-1,
0x1.4ap-1, 0x1.48p-1, 0x1.46p-1, 0x1.46p-1, 0x1.44p-1, 0x1.42p-1, 0x1.4p-1,
0x1.3ep-1, 0x1.3ep-1, 0x1.3cp-1, 0x1.3ap-1, 0x1.38p-1, 0x1.38p-1, 0x1.36p-1,
0x1.34p-1, 0x1.32p-1, 0x1.32p-1, 0x1.3p-1, 0x1.2ep-1, 0x1.2ep-1, 0x1.2cp-1,
0x1.2ap-1, 0x1.2ap-1, 0x1.28p-1, 0x1.26p-1, 0x1.26p-1, 0x1.24p-1, 0x1.22p-1,
0x1.22p-1, 0x1.2p-1, 0x1.1ep-1, 0x1.1ep-1, 0x1.1cp-1, 0x1.1cp-1, 0x1.1ap-1,
0x1.18p-1, 0x1.18p-1, 0x1.16p-1, 0x1.16p-1, 0x1.14p-1, 0x1.12p-1, 0x1.12p-1,
0x1.1p-1, 0x1.1p-1, 0x1.0ep-1, 0x1.0ep-1, 0x1.0cp-1, 0x1.0ap-1, 0x1.0ap-1,
0x1.08p-1, 0x1.08p-1, 0x1.06p-1, 0x1.06p-1, 0x1.04p-1, 0x1.04p-1, 0x1.02p-1,
0x1.02p-1, 0x1p-1,
};
// Generated by Sollya with:
// for i from 0 to 127 do {
// r = 2^-8 * nearestint( 2^8 * (1 + (i + 0.5)*2^(-7) - 2^(-15)) /
// ((1 + i*2^-7)*(1 + (i + 1)*2^-7)) );
// b = nearestint(log10(r)*2^43) * 2^-43;
// c = round(log10(r) - b, D, RN);
// print("{", -c, ",", -b, "},");
// };
// We replace LOG10_R[0] with log10(1.0) == 0.0
constexpr fputil::DoubleDouble LOG10_R[128] = {
{0.0, 0.0},
{-0x1.dfa6d47e47379p-45, 0x1.4f8205236p-8},
{-0x1.11bc6f2b9a3acp-45, 0x1.18b2dc8d3p-7},
{-0x1.685fc114e61bfp-46, 0x1.8a8c06bb2p-7},
{0x1.3c757d5b7376ap-45, 0x1.fd503c39p-7},
{0x1.f3cbd9c5111b1p-47, 0x1.3881a7b818p-6},
{-0x1.e22fab794a816p-45, 0x1.559bd2407p-6},
{0x1.421bab5f034a4p-45, 0x1.902c31d628p-6},
{0x1.fedb4b594a31bp-45, 0x1.cb38fccd88p-6},
{-0x1.c4e824b246c7fp-47, 0x1.0362241e64p-5},
{-0x1.df23ed2ebe477p-45, 0x1.125d0432ecp-5},
{-0x1.00c12f7a1b586p-47, 0x1.30838cdc3p-5},
{0x1.e5ff3439d368dp-46, 0x1.4eec0e2458p-5},
{0x1.2951bb9cd2fb7p-45, 0x1.5e3966b7e8p-5},
{0x1.fae96a708581ep-46, 0x1.7d070145f4p-5},
{0x1.badcf3d6e4566p-46, 0x1.9c197abfp-5},
{-0x1.b0197d2cb982ep-48, 0x1.abbcebd85p-5},
{-0x1.24b4a6b5ce4d4p-52, 0x1.cb38fccd8cp-5},
{-0x1.40bcd23c3e44cp-45, 0x1.db11ed766cp-5},
{-0x1.024e9d08ce301p-45, 0x1.fafa6d398p-5},
{0x1.77c3e779b9fcfp-48, 0x1.0585283764p-4},
{0x1.a9a57734f2038p-48, 0x1.15b11a094ap-4},
{-0x1.d227d61f9e88dp-45, 0x1.1dd5460c8cp-4},
{0x1.d288560689912p-53, 0x1.2e3a740b78p-4},
{-0x1.df5de49ddb16p-46, 0x1.367ba3aaa2p-4},
{0x1.5b873a39e56dcp-47, 0x1.471ba8a7dep-4},
{0x1.75da8a5871b9ap-45, 0x1.4f7aad9bbcp-4},
{0x1.ef5f3c10990f4p-45, 0x1.57e3d47c3ap-4},
{-0x1.02359c584ac3cp-45, 0x1.68d4eaf26ep-4},
{-0x1.41149840eaa65p-46, 0x1.715d0ce368p-4},
{-0x1.ff281b9601ce6p-46, 0x1.79efb57b1p-4},
{0x1.2b9da13d5c8cbp-47, 0x1.8b350364c6p-4},
{-0x1.80743406505e6p-48, 0x1.93e7de0fc4p-4},
{-0x1.76b169f6b4949p-49, 0x1.9ca5aa172ap-4},
{-0x1.bc8889d0fe1ap-47, 0x1.a56e8325f6p-4},
{-0x1.03ad4133e8c4cp-45, 0x1.b721cd1716p-4},
{0x1.72a4e1d198491p-46, 0x1.c00c776722p-4},
{-0x1.ddd18dedb6656p-45, 0x1.c902a19e66p-4},
{0x1.5e533080ecf32p-47, 0x1.d204698cb4p-4},
{0x1.7e865b8783768p-45, 0x1.db11ed766ap-4},
{0x1.5f7ef576ada0cp-45, 0x1.ed50a4a26ep-4},
{0x1.c9a3bd0891bccp-46, 0x1.f68216c9ccp-4},
{-0x1.ff229f20ed3d2p-46, 0x1.ffbfc2bbc8p-4},
{-0x1.6f30673aae7efp-45, 0x1.0484e4942bp-3},
{0x1.dae5ed5e3f34cp-45, 0x1.093025a199p-3},
{-0x1.3eea49e637bb3p-45, 0x1.0de1b56357p-3},
{0x1.82c6326f70b35p-46, 0x1.1299a4fb3ep-3},
{0x1.f04d633b79054p-45, 0x1.175805d158p-3},
{0x1.8b891b6d05a73p-48, 0x1.1c1ce9955cp-3},
{-0x1.35ca658049a0ap-51, 0x1.20e8624039p-3},
{0x1.ff081a4e81f0bp-45, 0x1.25ba8215afp-3},
{0x1.1e3f04f63ee01p-45, 0x1.2a935ba5f1p-3},
{-0x1.e1471e5cb397ep-45, 0x1.2f7301cf4fp-3},
{-0x1.5bd54fd6eb7d9p-45, 0x1.345987bfefp-3},
{0x1.fe93f791a7264p-46, 0x1.394700f795p-3},
{-0x1.8aebce3ec8738p-45, 0x1.3e3b814974p-3},
{0x1.b0722aa2559f2p-45, 0x1.43371cde07p-3},
{-0x1.bcb784188a058p-46, 0x1.4839e83507p-3},
{0x1.20ca9a2bcc728p-45, 0x1.4d43f8275ap-3},
{0x1.bb95ec8fd8f68p-45, 0x1.525561e925p-3},
{0x1.4e036062e2e73p-48, 0x1.576e3b0bdep-3},
{-0x1.e560b5e7a02b4p-51, 0x1.5c8e998073p-3},
{0x1.1e472c751a4cp-49, 0x1.61b6939983p-3},
{-0x1.1116ad28239bp-48, 0x1.66e6400da4p-3},
{0x1.9ad1d9e405fb9p-46, 0x1.6c1db5f9bbp-3},
{-0x1.41149840eaa65p-45, 0x1.715d0ce368p-3},
{0x1.bfb1de334b1cbp-45, 0x1.76a45cbb7ep-3},
{0x1.bfb1de334b1cbp-45, 0x1.76a45cbb7ep-3},
{-0x1.9fc01708d86e7p-48, 0x1.7bf3bde09ap-3},
{0x1.4adaf7fe992b6p-45, 0x1.814b4921bdp-3},
{-0x1.c0b434f5d2b7ap-46, 0x1.86ab17c10cp-3},
{0x1.4c7acd659652fp-45, 0x1.8c13437695p-3},
{0x1.3e99da1b485cfp-45, 0x1.9183e67339p-3},
{0x1.3e99da1b485cfp-45, 0x1.9183e67339p-3},
{-0x1.fb7d8e74e02ap-46, 0x1.96fd1b63ap-3},
{0x1.7c9690248757ep-46, 0x1.9c7efd734ap-3},
{-0x1.0cee0ed4ca7e9p-52, 0x1.a209a84fbdp-3},
{0x1.d924eb1fec6c5p-47, 0x1.a79d382bc2p-3},
{0x1.d924eb1fec6c5p-47, 0x1.a79d382bc2p-3},
{0x1.fe6eb5a4df1dfp-49, 0x1.ad39c9c2c6p-3},
{0x1.4c9cce6dd603bp-46, 0x1.b2df7a5c5p-3},
{-0x1.a01b9bb0ebf1bp-45, 0x1.b88e67cf98p-3},
{-0x1.a01b9bb0ebf1bp-45, 0x1.b88e67cf98p-3},
{0x1.2ee896e06dbe8p-45, 0x1.be46b08735p-3},
{-0x1.ff2381071d23ep-49, 0x1.c4087384f5p-3},
{-0x1.2f9fd61140aa6p-45, 0x1.c9d3d065c6p-3},
{-0x1.2f9fd61140aa6p-45, 0x1.c9d3d065c6p-3},
{-0x1.2386ad8b819b8p-45, 0x1.cfa8e765ccp-3},
{0x1.d63da3ac9f0d5p-45, 0x1.d587d96494p-3},
{0x1.d63da3ac9f0d5p-45, 0x1.d587d96494p-3},
{0x1.fcc16f3ba09cbp-45, 0x1.db70c7e96ep-3},
{-0x1.cc52c1ba2d838p-45, 0x1.e163d527e7p-3},
{-0x1.cc52c1ba2d838p-45, 0x1.e163d527e7p-3},
{0x1.f97877007c127p-46, 0x1.e76124046bp-3},
{0x1.fbd42fdc335fdp-47, 0x1.ed68d81919p-3},
{0x1.fbd42fdc335fdp-47, 0x1.ed68d81919p-3},
{-0x1.cbc0789055864p-45, 0x1.f37b15bab1p-3},
{0x1.2737df7f29668p-45, 0x1.f99801fdb7p-3},
{0x1.2737df7f29668p-45, 0x1.f99801fdb7p-3},
{-0x1.ff229f20ed3d2p-45, 0x1.ffbfc2bbc8p-3},
{0x1.00809c16ae3ecp-46, 0x1.02f93f4c87p-2},
{0x1.00809c16ae3ecp-46, 0x1.02f93f4c87p-2},
{-0x1.aa1358e87a6b4p-45, 0x1.06182e84fd8p-2},
{-0x1.aa1358e87a6b4p-45, 0x1.06182e84fd8p-2},
{-0x1.f171b2c5f2274p-48, 0x1.093cc32c91p-2},
{-0x1.42d6ae59e7d9cp-45, 0x1.0c6711d6acp-2},
{-0x1.42d6ae59e7d9cp-45, 0x1.0c6711d6acp-2},
{0x1.eac1871dbdbbfp-45, 0x1.0f972f87ffp-2},
{0x1.eac1871dbdbbfp-45, 0x1.0f972f87ffp-2},
{0x1.feed957c16a44p-46, 0x1.12cd31b9c98p-2},
{0x1.a6023a51d15b6p-46, 0x1.16092e5d3a8p-2},
{0x1.a6023a51d15b6p-46, 0x1.16092e5d3a8p-2},
{0x1.cefc5208422d8p-45, 0x1.194b3bdef68p-2},
{0x1.cefc5208422d8p-45, 0x1.194b3bdef68p-2},
{-0x1.1d4f1e8c2daffp-55, 0x1.1c93712abc8p-2},
{-0x1.1d4f1e8c2daffp-55, 0x1.1c93712abc8p-2},
{0x1.40eb9b53054c3p-46, 0x1.1fe1e5af2cp-2},
{0x1.9bbc8038401fcp-45, 0x1.2336b161b3p-2},
{0x1.9bbc8038401fcp-45, 0x1.2336b161b3p-2},
{0x1.09ca54daae9f9p-48, 0x1.2691ecc29fp-2},
{0x1.09ca54daae9f9p-48, 0x1.2691ecc29fp-2},
{0x1.2b528446968a4p-48, 0x1.29f3b0e1558p-2},
{0x1.2b528446968a4p-48, 0x1.29f3b0e1558p-2},
{-0x1.4548507c3dd04p-46, 0x1.2d5c1760b88p-2},
{-0x1.4548507c3dd04p-46, 0x1.2d5c1760b88p-2},
{-0x1.db59b99249f3ap-46, 0x1.30cb3a7bb38p-2},
{-0x1.db59b99249f3ap-46, 0x1.30cb3a7bb38p-2},
{0x1.fef311f12b358p-46, 0x1.34413509f78p-2},
};
constexpr double LOG10_R_ULP[2] = {0x1.0p-96, 0.0};
// Generated with Sollya:
// > P = fpminimax(log10(1 + x)/x, 6, [|D...|], [-2^-7; 2^-7], absolute);
// > dirtyinfnorm(log10(1 + x)/x - P, [-2^-7, 2^-7]);
// 0x1.9535684fb3064623001de9b13e6adf06355b5d75bp-57
constexpr double COEFFS[7] = {0x1.bcb7b1526e50ep-2, -0x1.bcb7b1526e53fp-3,
0x1.287a763700e4p-3, -0x1.bcb7b14641063p-4,
0x1.63c61abdf033fp-4, -0x1.28808b8a217fcp-4,
0x1.ffe99fc1908c6p-5};
constexpr double P_ERR = 0x1.0p-52;
// Number of extra range reduction steps.
constexpr size_t R_STEPS = 5;
constexpr size_t R_BITS = 4;
constexpr size_t R_SIZES = 1 << (R_BITS + 1);
// Generated by Sollya with:
// for i from 0 to 4 do {
// N = 11 + 4*i;
// print ("{");
// for j from -2^4 to 2^4 - 1 do {
// r = 2^(-N) * nearestint(2^(N) * ( 1 + (j + 0.5)*2^(-N) - 2^(-2*N-1)) /
// ((1 + j * 2^(-N)) * (1 + (j + 1)*2^(-N))));
// print(r, ",");
// };
// print("},");
// };
constexpr double RR[R_STEPS][R_SIZES] = {
{
0x1.02p0, 0x1.01ep0, 0x1.01cp0, 0x1.01ap0, 0x1.018p0, 0x1.016p0,
0x1.014p0, 0x1.012p0, 0x1.01p0, 0x1.00ep0, 0x1.00cp0, 0x1.00ap0,
0x1.008p0, 0x1.006p0, 0x1.004p0, 0x1p0, 0x1p0, 0x1.ffcp-1,
0x1.ff8p-1, 0x1.ff4p-1, 0x1.ffp-1, 0x1.fecp-1, 0x1.fe8p-1, 0x1.fe4p-1,
0x1.fep-1, 0x1.fdcp-1, 0x1.fd8p-1, 0x1.fd4p-1, 0x1.fdp-1, 0x1.fccp-1,
0x1.fc8p-1, 0x1.fc4p-1,
},
{
0x1.002p0, 0x1.001ep0, 0x1.001cp0, 0x1.001ap0, 0x1.0018p0,
0x1.0016p0, 0x1.0014p0, 0x1.0012p0, 0x1.001p0, 0x1.000ep0,
0x1.000cp0, 0x1.000ap0, 0x1.0008p0, 0x1.0006p0, 0x1.0004p0,
0x1p0, 0x1p0, 0x1.fffcp-1, 0x1.fff8p-1, 0x1.fff4p-1,
0x1.fffp-1, 0x1.ffecp-1, 0x1.ffe8p-1, 0x1.ffe4p-1, 0x1.ffep-1,
0x1.ffdcp-1, 0x1.ffd8p-1, 0x1.ffd4p-1, 0x1.ffdp-1, 0x1.ffccp-1,
0x1.ffc8p-1, 0x1.ffc4p-1,
},
{
0x1.0002p0, 0x1.0001ep0, 0x1.0001cp0, 0x1.0001ap0, 0x1.00018p0,
0x1.00016p0, 0x1.00014p0, 0x1.00012p0, 0x1.0001p0, 0x1.0000ep0,
0x1.0000cp0, 0x1.0000ap0, 0x1.00008p0, 0x1.00006p0, 0x1.00004p0,
0x1p0, 0x1p0, 0x1.ffffcp-1, 0x1.ffff8p-1, 0x1.ffff4p-1,
0x1.ffffp-1, 0x1.fffecp-1, 0x1.fffe8p-1, 0x1.fffe4p-1, 0x1.fffep-1,
0x1.fffdcp-1, 0x1.fffd8p-1, 0x1.fffd4p-1, 0x1.fffdp-1, 0x1.fffccp-1,
0x1.fffc8p-1, 0x1.fffc4p-1,
},
{
0x1.00002p0, 0x1.00001ep0, 0x1.00001cp0, 0x1.00001ap0,
0x1.000018p0, 0x1.000016p0, 0x1.000014p0, 0x1.000012p0,
0x1.00001p0, 0x1.00000ep0, 0x1.00000cp0, 0x1.00000ap0,
0x1.000008p0, 0x1.000006p0, 0x1.000004p0, 0x1p0,
0x1p0, 0x1.fffffcp-1, 0x1.fffff8p-1, 0x1.fffff4p-1,
0x1.fffffp-1, 0x1.ffffecp-1, 0x1.ffffe8p-1, 0x1.ffffe4p-1,
0x1.ffffep-1, 0x1.ffffdcp-1, 0x1.ffffd8p-1, 0x1.ffffd4p-1,
0x1.ffffdp-1, 0x1.ffffccp-1, 0x1.ffffc8p-1, 0x1.ffffc4p-1,
},
{
0x1.000002p0, 0x1.000001ep0, 0x1.000001cp0, 0x1.000001ap0,
0x1.0000018p0, 0x1.0000016p0, 0x1.0000014p0, 0x1.0000012p0,
0x1.000001p0, 0x1.000000ep0, 0x1.000000cp0, 0x1.000000ap0,
0x1.0000008p0, 0x1.0000006p0, 0x1.0000004p0, 0x1p0,
0x1p0, 0x1.ffffffcp-1, 0x1.ffffff8p-1, 0x1.ffffff4p-1,
0x1.ffffffp-1, 0x1.fffffecp-1, 0x1.fffffe8p-1, 0x1.fffffe4p-1,
0x1.fffffep-1, 0x1.fffffdcp-1, 0x1.fffffd8p-1, 0x1.fffffd4p-1,
0x1.fffffdp-1, 0x1.fffffccp-1, 0x1.fffffc8p-1, 0x1.fffffc4p-1,
},
};
// log10(2) with 192-bit prepcision generated by SageMath with:
// sage: (s, m, e) = RealField(192)(2).log10().sign_exponent_mantissa();
// sage: print("MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64), ",",
// hex((m >> 128) % 2^64), "})");
const Float192 LOG10_2(/*sign=*/false, /*exponent=*/-193, /*mantissa=*/
MType({0x26ad30c543d1f34a, 0x8f8959ac0b7c9178,
0x9a209a84fbcff798}));
// -log10(r) with 192-bit precision generated by SageMath with:
//
// for i in range(128):
// r = 2^-8 * round( 2^8 * (1 + (i + 1/2)*2^(-7) - 2^(-15)) / ((1 + i*2^-7)*(1
// + (i + 1)*2^-7)) ); s, m, e = RR(r).log10().sign_mantissa_exponent();
// print("{false,", e, ", MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64),
// ",", hex((m >> 128) % 2^64), "})},");
const Float192 LOG10_R_F192[128] = {
{false, 0, MType(0)},
{false, -199,
MType({0x3b7bb8a51a78c8bd, 0x6e321c6a7cdecc4, 0xa7c10291a88164ae})},
{false, -198,
MType({0xf4ba30d77042aac0, 0xa8cb8a86f6040a23, 0x8c596e4695dc8721})},
{false, -198,
MType({0x49fd078256b3ba8b, 0xeb19e4156cfa1bb7, 0xc546035d8e97a03e})},
{false, -198,
MType({0x3d964e8c5b12cf00, 0xb6e6ed36c9800088, 0xfea81e1c8278eafa})},
{false, -197,
MType({0x9c97a4a794669437, 0x714446c4f6a91d15, 0x9c40d3dc0c7cf2f6})},
{false, -197,
MType({0x1fc57458d58421ab, 0x86b57ea610c7db33, 0xaacde920361dd054})},
{false, -197,
MType({0x89be5385cc62fb89, 0x5f034a40e6a2f09c, 0xc81618eb15421bab})},
{false, -197,
MType({0x8b3be744561e443e, 0x594a31b2c5cc891b, 0xe59c7e66c5fedb4b})},
{false, -196,
MType({0xb1afe1d0add7d035, 0x69b7270237094316, 0x81b1120f31c762fb})},
{false, -196,
MType({0x6821aa1f743fb5b9, 0x68a0dc47567691c9, 0x892e821975106e09})},
{false, -196,
MType({0x1935f7436a3ea32c, 0x10bc94f44d216b49, 0x9841c66e17dfe7da})},
{false, -196,
MType({0x3dfba019afa885f9, 0xe74da327d8d80cb, 0xa77607122c797fcd})},
{false, -196,
MType({0x7d002c9ce2b6dff3, 0xce697dbaa00d4c7d, 0xaf1cb35bf494a8dd})},
{false, -196,
MType({0x9ff137c69b2cf640, 0x9c216079dcf0ea95, 0xbe8380a2fa7eba5a})},
{false, -196,
MType({0x94d5d9c17fa207c5, 0xf5b91598205b8144, 0xce0cbd5f806eb73c})},
{false, -196,
MType({0x8f995b90cbedc22d, 0x2d3467d253e2d1fb, 0xd5de75ec27e4fe68})},
{false, -196,
MType({0x8b3be744561e443e, 0x594a31b2c5cc891b, 0xe59c7e66c5fedb4b})},
{false, -196,
MType({0x97079feab7423d9b, 0xe1e0dda0b3d375a3, 0xed88f6bb355fa196})},
{false, -196,
MType({0xe86f834386695343, 0x7b98e7f593daf19e, 0xfd7d369cbf7ed8b1})},
{false, -195,
MType({0x5d4cae75ae8c48e7, 0x3bcdcfe7b23976cd, 0x82c2941bb20bbe1f})},
{false, -195,
MType({0x67047204db2989f7, 0xb9a7901be7521304, 0x8ad88d04a50d4d2b})},
{false, -195,
MType({0x829acbf3088a2849, 0x78185dcc37fda019, 0x8eeaa306458b760a})},
{false, -195,
MType({0x9d5d4ea0a7c8afb1, 0x1581a26448e2ac0e, 0x971d3a05bc0074a2})},
{false, -195,
MType({0xa078ffeb008a04d8, 0x6c449d409f883fe2, 0x9b3dd1d550c41443})},
{false, -195,
MType({0x2ede0aa760148d65, 0xa39e56dbb661c829, 0xa38dd453ef15b873})},
{false, -195,
MType({0x1716e6f9b3225217, 0x961c6e690d8879b4, 0xa7bd56cdde5d76a2})},
{false, -195,
MType({0x5bcb3818decefa22, 0x42643ced81ec14a, 0xabf1ea3e1d7bd7cf})},
{false, -195,
MType({0x164c684a1951204a, 0xe9ed4f101c5ef101, 0xb46a757936bf7298})},
{false, -195,
MType({0xcff88d82b3417f9b, 0xf7e2ab36f09e9013, 0xb8ae8671b3d7dd6c})},
{false, -195,
MType({0x8a86f0b2dfd799c2, 0x8d3fc63485e7ff12, 0xbcf7dabd87c01afc})},
{false, -195,
MType({0xf0e9563d52c1d533, 0x13d5c8cb231a53bd, 0xc59a81b26312b9da})},
{false, -195,
MType({0xa004d184e2faf2ea, 0x5fcd7d0ce937375e, 0xc9f3ef07e1f3fc5e})},
{false, -195,
MType({0x8a03e643ccfc8ec7, 0x58252dada9f06110, 0xce52d50b94fa253a})},
{false, -195,
MType({0xed0ad59454d36575, 0x62f01e5ff43708aa, 0xd2b74192fae43777})},
{false, -195,
MType({0x3af155936af69c0b, 0xb305ced1419fe924, 0xdb90e68b8abf14af})},
{false, -195,
MType({0x77f157444abbe2fc, 0x3a330921681e2481, 0xe0063bb3912e549c})},
{false, -195,
MType({0x693eca0a0148cc18, 0x849266a85513dc6d, 0xe48150cf32888b9c})},
{false, -195,
MType({0x73dac2c5ab4c8eda, 0x80ecf3266b4dcf4, 0xe90234c65a15e533})},
{false, -195,
MType({0x97079feab7423d9b, 0xe1e0dda0b3d375a3, 0xed88f6bb355fa196})},
{false, -195,
MType({0x22b519828722ce77, 0x5dab68307fedefcd, 0xf6a852513757dfbd})},
{false, -195,
MType({0x66278c914a763228, 0xa112379749fd91b8, 0xfb410b64e6393477})},
{false, -195,
MType({0x12b0b091e7b0299d, 0x1be2585c279c50a5, 0xffdfe15de3c01bac})},
{false, -194,
MType({0x3f42ceed6e05646f, 0x18aa302171017dcb, 0x8242724a155219f3})},
{false, -194,
MType({0xd146a4518aa0076f, 0xabc7e698502d43bf, 0x849812d0ccbb5cbd})},
{false, -194,
MType({0x34708f4b01b4e73f, 0xc339089a51663370, 0x86f0dab1ab5822b6})},
{false, -194,
MType({0x5e3e593143a3f512, 0x26f70b34ce5cf201, 0x894cd27d9f182c63})},
{false, -194,
MType({0xb5f326771e3d87a9, 0x676f20a87ab433de, 0x8bac02e8ac3e09ac})},
{false, -194,
MType({0x633dc078567bc8d9, 0x6db4169cc4b83bc3, 0x8e0e74caae062e24})},
{false, -194,
MType({0x662d0eb20c51da4f, 0xcd3fdb2fad0d1fd6, 0x907431201c7f651a})},
{false, -194,
MType({0x7fed29520fdf89d7, 0x49d03e163250d1d4, 0x92dd410ad7bfe103})},
{false, -194,
MType({0xed5106d40e889904, 0x9ec7dc02d5e723b8, 0x9549add2f8a3c7e0})},
{false, -194,
MType({0xcdbe2ebc07714f34, 0x34698d03a5442572, 0x97b980e7a743d71c})},
{false, -194,
MType({0x1e9782e43af2d3d7, 0x522904d1e47f3de, 0x9a2cc3dff7548556})},
{false, -194,
MType({0xfabfd5317e9bdd56, 0x791a72646c87b975, 0x9ca3807bca9fe93f})},
{false, -194,
MType({0x2dc99e57977305af, 0x3826f190d655d736, 0x9f1dc0a4b9cea286})},
{false, -194,
MType({0x5a4dfeb5b0db1f24, 0x544ab3e48199b299, 0xa19b8e6f03b60e45})},
{false, -194,
MType({0x2888ece2bf37b7e9, 0xbe775fa82961114e, 0xa41cf41a83643487})},
{false, -194,
MType({0xef4915fda0982302, 0x45798e5019e6c081, 0xa6a1fc13ad241953})},
{false, -194,
MType({0x33a92c4634bdd6c, 0x91fb1ed0cdc4d1fb, 0xa92ab0f492b772bd})},
{false, -194,
MType({0xe89863702c85cccb, 0x818b8b9cbbd17b71, 0xabb71d85ef05380d})},
{false, -194,
MType({0x482ab24ae8fdf5a4, 0xa50c2fea60c5b3b2, 0xae474cc0397f0d4f})},
{false, -194,
MType({0x682d5b55e9594eb3, 0x58ea34980ad8b720, 0xb0db49ccc1823c8e})},
{false, -194,
MType({0xae8dceb953c096c0, 0x4b5f71941be508a3, 0xb3732006d1fbbba5})},
{false, -194,
MType({0xfc1396a39c34fef3, 0x9e405fb8bcb1ff1d, 0xb60edafcdd99ad1d})},
{false, -194,
MType({0xcff88d82b3417f9b, 0xf7e2ab36f09e9013, 0xb8ae8671b3d7dd6c})},
{false, -194,
MType({0x6f1a4043a1a8a435, 0xc669639640c305bb, 0xbb522e5dbf37f63b})},
{false, -194,
MType({0x6f1a4043a1a8a435, 0xc669639640c305bb, 0xbb522e5dbf37f63b})},
{false, -194,
MType({0x6c4433809b0babe7, 0xa3dc9e464e98764b, 0xbdf9def04cf980ff})},
{false, -194,
MType({0x38a1d9b9b9370d6d, 0xffd3256b59fa9c59, 0xc0a5a490dea95b5e})},
{false, -194,
MType({0x60b092e62b5beb8a, 0xb0a2d48672a051a5, 0xc3558be085e3f4bc})},
{false, -194,
MType({0x1065867f127536e0, 0xacb2ca5d4ca1c10e, 0xc609a1bb4aa98f59})},
{false, -194,
MType({0x8030cfce4646062f, 0x43690b9e3cde0d01, 0xc8c1f3399ca7d33b})},
{false, -194,
MType({0x8030cfce4646062f, 0x43690b9e3cde0d01, 0xc8c1f3399ca7d33b})},
{false, -194,
MType({0xd806ff9947bc6ca7, 0x18b1fd60383f7e59, 0xcb7e8db1cfe04827})},
{false, -194,
MType({0x65af114cbdb0193e, 0x248757e5f45af3d, 0xce3f7eb9a517c969})},
{false, -194,
MType({0x3569862a1e8f9a4c, 0x7c4acd605be48bc1, 0xd104d427de7fbcc4})},
{false, -194,
MType({0x94e3cbc5cd693dda, 0x58ff63629a92652c, 0xd3ce9c15e10ec927})},
{false, -194,
MType({0x94e3cbc5cd693dda, 0x58ff63629a92652c, 0xd3ce9c15e10ec927})},
{false, -194,
MType({0x1e0a73c970dbbf33, 0x6b49be3bd8c89f10, 0xd69ce4e16303fcdd})},
{false, -194,
MType({0x59899d5040e21558, 0xe6dd603a881e9060, 0xd96fbd2e2814c9cc})},
{false, -194,
MType({0x71549f0a4d78d49c, 0x89e281c98c1d705c, 0xdc4733e7cbcbfc8c})},
{false, -194,
MType({0x71549f0a4d78d49c, 0x89e281c98c1d705c, 0xdc4733e7cbcbfc8c})},
{false, -194,
MType({0xf4eee5981334e57, 0xdc0db7cf0cce9f32, 0xdf2358439aa5dd12})},
{false, -194,
MType({0xd50afdf84e68b269, 0xfdf1c5b846db9dea, 0xe20439c27a7c01b8})},
{false, -194,
MType({0xd95ac10b65558e44, 0x3dd7eab48869c401, 0xe4e9e832e2da0c05})},
{false, -194,
MType({0xd95ac10b65558e44, 0x3dd7eab48869c401, 0xe4e9e832e2da0c05})},
{false, -194,
MType({0xe9377fb1f3b453b6, 0x4e8fcc900b41daee, 0xe7d473b2e5db8f2a})},
{false, -194,
MType({0xe772954c39d07f3b, 0x7593e1a9e9173599, 0xeac3ecb24a3ac7b4})},
{false, -194,
MType({0xe772954c39d07f3b, 0x7593e1a9e9173599, 0xeac3ecb24a3ac7b4})},
{false, -194,
MType({0xa6d10312a95362d4, 0xe7741396b49e1ce4, 0xedb863f4b73f982d})},
{false, -194,
MType({0xd0d55aebaeb5abfc, 0xc8ba4f8f47b85a5b, 0xf0b1ea93f34675a7})},
{false, -194,
MType({0xd0d55aebaeb5abfc, 0xc8ba4f8f47b85a5b, 0xf0b1ea93f34675a7})},
{false, -194,
MType({0x7e1dea1275662695, 0x7007c1276821b705, 0xf3b09202359f9787})},
{false, -194,
MType({0x54dc283e4f79339c, 0x7ee19afe6db7e324, 0xf6b46c0c8c8fdea1})},
{false, -194,
MType({0x54dc283e4f79339c, 0x7ee19afe6db7e324, 0xf6b46c0c8c8fdea1})},
{false, -194,
MType({0xf7844244016096c0, 0xedf54f37f6d4041f, 0xf9bd8add584687f0})},
{false, -194,
MType({0x94a99151573d5249, 0xefe52ccf03e7dee0, 0xfccc00fedba4e6fb})},
{false, -194,
MType({0x94a99151573d5249, 0xefe52ccf03e7dee0, 0xfccc00fedba4e6fb})},
{false, -194,
MType({0x12b0b091e7b0299d, 0x1be2585c279c50a5, 0xffdfe15de3c01bac})},
{false, -193,
MType({0xeba2ae5f7d1c7168, 0xe0b571f5c91b0445, 0x817c9fa643880404})},
{false, -193,
MType({0xeba2ae5f7d1c7168, 0xe0b571f5c91b0445, 0x817c9fa643880404})},
{false, -193,
MType({0x2db8874aa1eb0c3c, 0x7178594bef2def59, 0x830c17427ea55eca})},
{false, -193,
MType({0x2db8874aa1eb0c3c, 0x7178594bef2def59, 0x830c17427ea55eca})},
{false, -193,
MType({0xf3cb58bd1bbe04f0, 0x9a741bb171158d29, 0x849e6196487c1d1c})},
{false, -193,
MType({0xd95b712663014da, 0x1a618264446cb495, 0x863388eb55ebd295})},
{false, -193,
MType({0xd95b712663014da, 0x1a618264446cb495, 0x863388eb55ebd295})},
{false, -193,
MType({0x2e66a10dfdce751, 0x71dbdbbec51d7657, 0x87cb97c3ff9eac18})},
{false, -193,
MType({0x2e66a10dfdce751, 0x71dbdbbec51d7657, 0x87cb97c3ff9eac18})},
{false, -193,
MType({0x84a2c0cd81dbcf53, 0xabe0b522230f7d13, 0x896698dce4cff76c})},
{false, -193,
MType({0xae1f8a1def2acf5a, 0xd28e8adafea703b3, 0x8b04972e9d4d3011})},
{false, -193,
MType({0xae1f8a1def2acf5a, 0xd28e8adafea703b3, 0x8b04972e9d4d3011})},
{false, -193,
MType({0x8a02390202a4a59d, 0x208422d83be34b26, 0x8ca59def7b5cefc5})},
{false, -193,
MType({0x8a02390202a4a59d, 0x208422d83be34b26, 0x8ca59def7b5cefc5})},
{false, -193,
MType({0x420c16bd3939f912, 0xc385cf49402af0e4, 0x8e49b8955e3ffb8a})},
{false, -193,
MType({0x420c16bd3939f912, 0xc385cf49402af0e4, 0x8e49b8955e3ffb8a})},
{false, -193,
MType({0x8a1754a1ee7c990, 0xda982a614e12c6dd, 0x8ff0f2d7960a075c})},
{false, -193,
MType({0xe77cb3d650c2718e, 0x38401fc1c1b5c2b, 0x919b58b0d999bbc8})},
{false, -193,
MType({0xe77cb3d650c2718e, 0x38401fc1c1b5c2b, 0x919b58b0d999bbc8})},
{false, -193,
MType({0x3c50b7234a381be8, 0xa9b55d3f16da746a, 0x9348f6614f821394})},
{false, -193,
MType({0x3c50b7234a381be8, 0xa9b55d3f16da746a, 0x9348f6614f821394})},
{false, -193,
MType({0xd31cd763e50a0231, 0x88d2d1473d4f7f4, 0x94f9d870aac256a5})},
{false, -193,
MType({0xd31cd763e50a0231, 0x88d2d1473d4f7f4, 0x94f9d870aac256a5})},
{false, -193,
MType({0x8eb2e675bc182d0d, 0x7c1e117dea19e9e5, 0x96ae0bb05c35d5bd})},
{false, -193,
MType({0x8eb2e675bc182d0d, 0x7c1e117dea19e9e5, 0x96ae0bb05c35d5bd})},
{false, -193,
MType({0x78c2d9cf6e98b1c1, 0x336db0630f536fb9, 0x98659d3dd9b12532})},
{false, -193,
MType({0x78c2d9cf6e98b1c1, 0x336db0630f536fb9, 0x98659d3dd9b12532})},
{false, -193,
MType({0x26ad30c543d1f34a, 0x8f8959ac0b7c9178, 0x9a209a84fbcff798})},
};
// -log10(r) for further range reduction steps, generated by SageMath with:
//
// RR = RealField(192);
// for i in range(5):
// N = 11 + 4*i;
// print("{");
// for j in range(-2^4, 2^4):
// r = 2^(-N) * round(2^(N) * ( 1 + (j + 0.5)*2^(-N) - 2^(-2*N-1) ) / ((1 +
// j * 2^(-N)) * (1 + (j + 1)*2^(-N)))); a = -RR(r).log10(); if j in [0,
// -1]:
// r = 1; a = RR(0);
// s, m, e = a.sign_mantissa_exponent()
// sgn = "{false," if s == 1 else "{true,";
// print(sgn, e, ", MType({", hex(m % 2^64), ",", hex((m >> 64) % 2^64),
// ",", hex((m >> 128) % 2^64), "})},");
// print("},");
const Float192 LOG10_RR[R_STEPS][R_SIZES] = {
{
{true, -200,
MType({0xf52b7aea9ca0c476, 0xdd4a47e1490df56, 0xdd7ea3910f69332e})},
{true, -200,
MType({0x11c54af8b7b5ac0, 0xbe14368ead9df21c, 0xcfb39f5a164f371c})},
{true, -200,
MType({0xdc8b8e1f46c98b22, 0xb46a6050fcd513ca, 0xc1e6e4dcf45e0ee0})},
{true, -200,
MType({0x10c91fa8440cf6a8, 0x799e2aecea385841, 0xb41873accfbaba29})},
{true, -200,
MType({0xefff2920f96d3591, 0x3440eb8005ad171b, 0xa6484b5ca5f7eaea})},
{true, -200,
MType({0xaa51f382c80587cb, 0x21e85464267c47a4, 0x98766b7f4c01d932})},
{true, -200,
MType({0xaf1454629688e40a, 0xee383c5f131898c4, 0x8aa2d3a76e0a0a79})},
{true, -201,
MType({0xf128ae4671df433, 0x15022fc12f1747d4, 0xf99b06cf1ee618b9})},
{true, -201,
MType({0x511d8d64285691ec, 0x411e70646c13ae93, 0xddecf4a415784564})},
{true, -201,
MType({0xd9e8330f453fac1c, 0x662bb5e078c532bf, 0xc23b6ff222d9d207})},
{true, -201,
MType({0xa9d37fde93b3c989, 0xebcfc062f09deb23, 0xa68677dd5801ce9b})},
{true, -201,
MType({0xe9eadfa1556bcaec, 0xbfdd2c7f388fc013, 0x8ace0b8973a63413})},
{true, -202,
MType({0x743a88400b3b04bc, 0xc21595e745f1fa15, 0xde245433c425b5c5})},
{true, -202,
MType({0xf50e04c0bd73e444, 0x663cf2b27e8f1ffa, 0xa6a5a5637a00afdc})},
{true, -203,
MType({0xf860837f66c35a6a, 0xba0324530edaa03e, 0xde4011cf2daaff31})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, -204,
MType({0x8319f4e45b099e86, 0x20fa8a73c585f633, 0xde69bf8f58005dfc})},
{false, -203,
MType({0x58e9d6dee00a8fc3, 0x35e3298f4bb2aa0a, 0xde77a8c714b08d28})},
{false, -202,
MType({0xc6b4c904be2cbf01, 0xa1e66c6203725d4f, 0xa6e42f3ccf49959d})},
{false, -202,
MType({0xebc8b4278d3f93e9, 0x5ad3480ccfbe988f, 0xde93822dfe812587})},
{false, -201,
MType({0x9e52f5c983d9caa4, 0xe5f7c52cdf119d4, 0x8b24e77b0cb60a84})},
{false, -201,
MType({0x9a29985625bf9936, 0x84e42dd54c2ff44e, 0xa7038baa64c5cd39})},
{false, -201,
MType({0x15c143896f774901, 0xde013ef4ba5abcfa, 0xc2e5ae853064a921})},
{false, -201,
MType({0x8c6b1043496971e, 0x8f6c475cc382401c, 0xdecb50ebece5e146})},
{false, -201,
MType({0x95457befdc15d5a7, 0x64a67faf3f1c74c2, 0xfab473bf6c2589b9})},
{false, -200,
MType({0x825120e3dbb4f086, 0x341aa1df5e8fab33, 0x8b508bf06a597f35})},
{false, -200,
MType({0xc69ade976589376a, 0x8b19159a7036ac6c, 0x99489f18d0fdba55})},
{false, -200,
MType({0x132b76e2a5ed5ce1, 0x7b8b2852580ae3e2, 0xa74273c9d23a53b9})},
{false, -200,
MType({0x6f0c9495185af64, 0xe9ad89ebfeaf5a02, 0xb53e0a7480e3cf3c})},
{false, -200,
MType({0x3dc11a3e41e693ee, 0xe12996e107931f4b, 0xc33b638a1a7dc81d})},
{false, -200,
MType({0x2ce27c62f7c5d30e, 0xb43caf8e7b891066, 0xd13a7f7c07506f7d})},
},
{
{true, -204,
MType({0xea57bded8e15fa61, 0x634cea6750617a91, 0xde4df4140b42822f})},
{true, -204,
MType({0x8a2e8bb53b241b41, 0x9185547f24195fab, 0xd069e52741db93b6})},
{true, -204,
MType({0x265b41c72630e265, 0xfbd7e970aef9dbb8, 0xc285ba757feb2781})},
{true, -204,
MType({0xd2e480f950aea396, 0xe820a1a6f319285a, 0xb4a173fe56691147})},
{true, -204,
MType({0xb3300685a98f7031, 0x9aedc1c1ba7d0694, 0xa6bd11c1564a8ace})},
{true, -204,
MType({0x407aba2445f75697, 0x89de4989587519ca, 0x98d893be108233d7})},
{true, -204,
MType({0xbe4f7e3cbefc71ce, 0x8d306ba207233c43, 0x8af3f9f41600120a})},
{true, -205,
MType({0x5449f3ae334e8294, 0x2100088006aeff92, 0xfa1e88c5ef6321c2})},
{true, -205,
MType({0x2245be0033c6dafe, 0x856a0a3a00fcf3c1, 0xde54e6148d030322})},
{true, -205,
MType({0xf8b7f31304662b13, 0x8a4399de0be1c455, 0xc28b0bd326b035f2})},
{true, -205,
MType({0x5abacdedb1dc0b2c, 0xb3a2c1407cf6d38d, 0xa6c0fa00de35f314})},
{true, -205,
MType({0x874695793b5a312e, 0x220af90d74ea17c7, 0x8af6b09cd55a3e68})},
{true, -206,
MType({0x70a9c05700a3f2c5, 0xd791cf6a70c3a504, 0xde585f4c5bbbcd3d})},
{true, -206,
MType({0x1f465fbffe379fdf, 0xe95b52a2781c25e3, 0xa6c2ee3812f90a28})},
{true, -207,
MType({0x35af2ed67a089e4, 0x10a633f2c4a8ea22, 0xde5a1bf627b1f68f})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, -208,
MType({0xe0faf0a837063878, 0x140a2bff40e0d66f, 0xde5cb706384ddbaf})},
{false, -207,
MType({0xd660ff616f334614, 0xed4a68e5e6e83dde, 0xde5d95658a729eab})},
{false, -206,
MType({0x542eeb05597ecdea, 0x7c9dc000df79edde, 0xa6c6d6d56238dd6e})},
{false, -206,
MType({0x5f8e2902ab25d093, 0x3281f1872cdbee94, 0xde5f522b21e3e25a})},
{false, -205,
MType({0xba7cb3452f07c347, 0x292ef5f6fbe96c00, 0x8afc1e5ae08b4643})},
{false, -205,
MType({0xde1cc00ce5b7249c, 0xf1466edaa96e356d, 0xa6c8cb3b7e5bbbfd})},
{false, -205,
MType({0xa521ef720e08d8fe, 0x936a112cb6265f7e, 0xc295afb848dbd759})},
{false, -205,
MType({0x1d1317f34a2fba9e, 0x8a607fd695dfc3d9, 0xde62cbd21e895473})},
{false, -205,
MType({0x9c74fa60d5502c53, 0x65a49326981bdf3f, 0xfa301f89dde726b4})},
{false, -204,
MType({0x4295ea162700a15d, 0xc36b8713ceefe2de, 0x8afed57032bebc7c})},
{false, -204,
MType({0x10e84795ee07f1c, 0x7068ee550f5ffb37, 0x98e5b6eb49ecd6df})},
{false, -204,
MType({0xf71013dc630dfafa, 0x5c2e76c953e3e3e5, 0xa6ccb436a3c72fa4})},
{false, -204,
MType({0x61fc137c9dce9644, 0xb3425e818888a098, 0xb4b3cd52af99afe6})},
{false, -204,
MType({0xe923dcabcfbadd8e, 0x8e4950fa5c943bbe, 0xc29b023fdcb2dccf})},
{false, -204,
MType({0xd941d748e2a23bcc, 0xd4d3796d93d6750b, 0xd08252fe9a63d7aa})},
},
{
{true, -208,
MType({0x22058fdb723bb37, 0xce31a0d27359396f, 0xde5afa4e86f7f3ee})},
{true, -208,
MType({0x2e22d8a1e7a512a1, 0x9f63aaa563e9a399, 0xd07557b0de924142})},
{true, -208,
MType({0xd1fffbb5f7c92f10, 0x67e63bbe1405c20c, 0xc28fb35684fedaab})},
{true, -208,
MType({0x2473a71714b9ef53, 0x5440d7a131392da8, 0xb4aa0d3f79ce9499})},
{true, -208,
MType({0xdec055ae33e59ad5, 0xe0e635a681d259e1, 0xa6c4656bbc924352})},
{true, -208,
MType({0x953365af70272280, 0xda1f690c752fdefe, 0x98debbdb4cdabaf4})},
{true, -208,
MType({0xc6c161d92abe9aa4, 0x5bf708fead2b177f, 0x8af9108e2a38cf72})},
{true, -209,
MType({0x78732843e6506315, 0xa448b11f012c975c, 0xfa26c708a87aa929})},
{true, -209,
MType({0x863d53d84705d0ae, 0xefecdd48ed894c31, 0xde5b697b94f23bf7})},
{true, -209,
MType({0x87704029104c21a9, 0xb07ebbab782457af, 0xc290087518f9fe3b})},
{true, -209,
MType({0x4adb1df9649c50c1, 0x9a7eb8811ec3e6bb, 0xa6c4a3f533b3968d})},
{true, -209,
MType({0xcfe3880042e86013, 0x11fd6995111a926, 0x8af93bfbe440ab33})},
{true, -210,
MType({0x2cbf7a2e7b9cbcf0, 0xac3bfd925e6b33e1, 0xde5ba1125385c43b})},
{true, -210,
MType({0x4d418d19762cbd66, 0x53289e84744549cb, 0xa6c4c33a06b7c1d9})},
{true, -211,
MType({0x4c3b3ad870a5dd12, 0xa74dab3bd6067bc7, 0xde5bbcddc0b533aa})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, -212,
MType({0x29423084899fc9b9, 0xbee710a5ace7c8d4, 0xde5be68ef5db7f99})},
{false, -211,
MType({0x1c57cf04091e59cc, 0x7b644b13993cf4ef, 0xde5bf474b6df8331})},
{false, -210,
MType({0xd259a65cb34a4a8a, 0x64136e97019d0a3a, 0xa6c501c3dba75dc2})},
{false, -210,
MType({0x6f497708cc18c4d8, 0xa2912e03afc8cc28, 0xde5c10403fda6db5})},
{false, -209,
MType({0x865e042dd50e1d3a, 0xbf5dccd967504856, 0x8af992d7c4e2d5b5})},
{false, -209,
MType({0xe1702b0f17301f13, 0xe22978efa7a9629f, 0xa6c52108dd92e8c1})},
{false, -209,
MType({0x74a88aedf1375fb4, 0x82cd723bf1524680, 0xc290b2b36adbcda2})},
{false, -209,
MType({0xa923a00c40053158, 0x4dd6f8576e9188b7, 0xde5c47d76d9be24e})},
{false, -209,
MType({0x923113e7e6eec47e, 0x5368655f1ce3110a, 0xfa27e074e6b1850f})},
{false, -208,
MType({0xee64cf0bae6dd4c9, 0x83b16ff7eecace8b, 0x8af9be45eb7d8a41})},
{false, -208,
MType({0x782dbb6881f2d3d, 0x20c8069e4ae400de, 0x98df8e0e1fab77cd})},
{false, -208,
MType({0xff9e92ff9157f7ff, 0x9ee5e19f2eecdb54, 0xa6c55f931051bacc})},
{false, -208,
MType({0x4260c63c04224e80, 0xce16dd7f60311bf6, 0xb4ab32d4bddf830b})},
{false, -208,
MType({0xbf40b2a151280765, 0x3099a17e0461648b, 0xc29107d328c40080})},
{false, -208,
MType({0xb32695665b8f1329, 0xfaf478d3d0b312ff, 0xd076de8e516e6348})},
},
{
{true, -212,
MType({0xdaf98ea8088ccd55, 0x19be3fabd93832c4, 0xde5bcac37ac6587d})},
{true, -212,
MType({0x92832d76e51450a5, 0x88a86b30027dc8a5, 0xd0760ee7b9136936})},
{true, -212,
MType({0xdd67de1487eb3913, 0xc61c028b8f230f14, 0xc29052f02bebe878})},
{true, -212,
MType({0x8b5c1ba51e27536f, 0xb8cfce8fa982ea7, 0xb4aa96dcd34f6716})},
{true, -212,
MType({0xe1a1b61dba0d85b7, 0x8fd43f0c9ce444d2, 0xa6c4daadaf3d75e0})},
{true, -212,
MType({0x60e206e644f9e560, 0x872f9b3fd2141246, 0x98df1e62bfb5a5aa})},
{true, -212,
MType({0xeef14d02362d0513, 0x2341d13c21a7d8cf, 0x8af961fc04b78746})},
{true, -213,
MType({0xb56b5c514a13e254, 0x26251c2ccc00d194, 0xfa274af2fc85570b})},
{true, -213,
MType({0x60d8b357006a3fed, 0x61cd853e796bc2c, 0xde5bd1b658ad4676})},
{true, -213,
MType({0xa90e9309cbc1b814, 0x3a0de60782dd1939, 0xc29058421de5fe71})},
{true, -213,
MType({0xb6bc3ab198cec5a2, 0x10652e9b1ce538bb, 0xa6c4de964c2ea0a1})},
{true, -213,
MType({0x27d91560ba3c3289, 0xd259757215d9aa0d, 0x8af964b2e3864ea9})},
{true, -214,
MType({0xef6374ae583f4c18, 0x87d6afabfba0644e, 0xde5bd52fc7d8545f})},
{true, -214,
MType({0x4bf40373e818d7f5, 0x47ca9999c73441f5, 0xa6c4e08a9abea9ae})},
{true, -215,
MType({0x256483d247de2b54, 0xaf6e93be8e4c1764, 0xde5bd6ec7f7bc110})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, -216,
MType({0x5f6593112c03bf0b, 0xd5c0ebcf0fa0221e, 0xde5bd98793024346})},
{false, -215,
MType({0x11b7b6679d212c54, 0x6d0063a923dd0cc6, 0xde5bda65eede65ed})},
{false, -214,
MType({0x6079e21db1a5b651, 0xc424c8976e2ebcfa, 0xa6c4e473380da326})},
{false, -214,
MType({0x8a7f3f059746be25, 0xa9bf32001043629c, 0xde5bdc22a69d9e19})},
{false, -213,
MType({0xc8dbe969fc9bb1cd, 0x9e8e77fa22e00248, 0x8af96a20a1907043})},
{false, -213,
MType({0xe129bb635500e8db, 0xaf3be9e7aa56d682, 0xa6c4e66786cc9393})},
{false, -213,
MType({0xea8e87193d810fa0, 0xfc3aff5aaf056566, 0xc29062e603041758})},
{false, -213,
MType({0x254d7de51cd5637d, 0x8014f0f360272d82, 0xde5bdf9c1637d9ef})},
{false, -213,
MType({0xe9795a136f5856c, 0x3a891f89bcf72a40, 0xfa275c89c068b9b3})},
{false, -212,
MType({0x3c5338931e9d2409, 0x18468a2ba2fa5549, 0x8af96cd780cbca80})},
{false, -212,
MType({0x3ab6a381bda506d5, 0x362640905714e443, 0x98df2b85ece2a519})},
{false, -212,
MType({0x13b57109141a8764, 0xfe94a02fc639c0e3, 0xa6c4ea5024795bd2})},
{false, -212,
MType({0xc613ec4aff62b164, 0x7bddaab60665b883, 0xb4aaa93627905ddb})},
{false, -212,
MType({0x29f60bd76dda9de7, 0xbae8765350c99434, 0xc2906837f6281a60})},
{false, -212,
MType({0x6d1b70f85e1ebdd6, 0xcb372dd0da70965b, 0xd076275590410090})},
},
{
{true, -216,
MType({0xcf260846db91386b, 0x81645201b36e17b9, 0xde5bd7cadb50f0d8})},
{true, -216,
MType({0x8f96908bc0be4701, 0xcdea7b1a0dc7ad41, 0xd0761a5b34fd720c})},
{true, -216,
MType({0x71e2e36ca8d985f3, 0x5ce6604b0911c5ed, 0xc2905ce9d1f24872})},
{true, -216,
MType({0xdaf377e6a6fc292b, 0x6e0982bd8d96ee, 0xb4aa9f76b22f739a})},
{true, -216,
MType({0xf97df5dd10eab18e, 0x8a9754fe0c0073a6, 0xa6c4e201d5b4f314})},
{true, -216,
MType({0x260badb439a3192f, 0xcd77f7489d9ef50b, 0x98df248b3c82c672})},
{true, -216,
MType({0x35f8aeb949a552f2, 0x9b257b3ce3f82109, 0x8af96712e698ed45})},
{true, -217,
MType({0x82d6f89adeda85fe, 0x8b6a840831c123d7, 0xfa275331a7eece3b})},
{true, -217,
MType({0xb88ac395452a99d1, 0x3e79062c7cbb3b7c, 0xde5bd83a093c6718})},
{true, -217,
MType({0x8416ca30683117d1, 0xf3a098743d20fb63, 0xc2905d3ef11aa442})},
{true, -217,
MType({0xe98fafcacc5781ed, 0x4f0b030a9742eaff, 0xa6c4e2405f8984dd})},
{true, -217,
MType({0x9dc90f008c47f3e0, 0xf4e1bab83f55f5a0, 0x8af9673e54890808})},
{true, -218,
MType({0x1ff62cfc0b87b23a, 0x129bc1c6f293726e, 0xde5bd871a03259cf})},
{true, -218,
MType({0xb2b2c5c7e8e3dd1b, 0x60f08720313daa3e, 0xa6c4e25fa473e535})},
{true, -219,
MType({0x4a6c56a5be458d59, 0x3a25757e00f4e3a0, 0xde5bd88d6bad6110})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, 0, MType({0x0, 0x0, 0x0})},
{false, -220,
MType({0x3f16581bde8d9c4a, 0x225916c2b3f33c90, 0xde5bd8b71ce5fd51})},
{false, -219,
MType({0x8d1575503bb9c7fd, 0x44397830931fddc, 0xde5bd8c502a38b5e})},
{false, -218,
MType({0x91d4a95df896a79f, 0xe454dec82bde52e4, 0xa6c4e29e2e48d4cc})},
{false, -218,
MType({0x242c88441091cf56, 0xa6e2721f2afc3cce, 0xde5bd8e0ce1eae6a})},
{false, -217,
MType({0x9b6ef45b7cbebc03, 0x80bf0ff2f6cd9f92, 0x8af967953069aa22})},
{false, -217,
MType({0x7fe96798a2b597a7, 0x55ee1480619827c4, 0xa6c4e2bd7333640c})},
{false, -217,
MType({0xedc6a1463ce32849, 0x2ed8ba8c6fa81c97, 0xc2905de92f6c85d1})},
{false, -217,
MType({0xd59f31668df9feb7, 0x6759c94e2d017fac, 0xde5bd9186515104f})},
{false, -217,
MType({0x337d83d075e0c1ab, 0x5b4c5b5f180b9fe1, 0xfa27544b142d0465})},
{false, -216,
MType({0x390379ad9871c8a7, 0xb345ef5d90dd6545, 0x8af967c09e5a3178})},
{false, -216,
MType({0x73d9d08521f9533d, 0xf27a0a6056dcfe57, 0x98df255d6f559668})},
{false, -216,
MType({0xb70a87a69080a500, 0x99308918494a4a1f, 0xa6c4e2fbfd08b172})},
{false, -216,
MType({0x8995c0b3074c289d, 0xd5579f970cf000a8, 0xb4aaa09c47738304})},
{false, -216,
MType({0x72650f85eed7af38, 0xd4ddab9f8032bbab, 0xc2905e3e4e960b8e})},
{false, -216,
MType({0xeb04ee2cfc701a37, 0xc5b134a5bb25cf2d, 0xd0761be212704b7f})},
},
};
// > P = fpminimax(log10(1 + x)/x, 4, [|192...|], [-2^-27, 2^-27]);
// > P;
// > dirtyinfnorm(log10(1 + x)/x - P, [-2^-27, 2^-27]);
// 0x1.287a7...p-143
const Float192 BIG_COEFFS[5]{
{false, -194,
MType({0x1fc14ee0c0158d73, 0x762ec7601912bf70, 0x58f189dd49436234})},
{true, -195,
MType({0x8f42a80e947f6357, 0x67836140b941fe04, 0xde5bd8a93728747a})},
{false, -194,
MType({0xf6f00690b1fa8ba9, 0x78e7c71fc8ca2d3a, 0x943d3b1b7a1af663})},
{true, -191,
MType({0x255a69358264a2d1, 0xa6ab7555f5a64d33, 0x1bcb7b1526e50e32})},
{false, -193,
MType({0x3ee3460246fdf301, 0x355baaafad33dc32, 0xde5bd8a937287195})},
};
// Reuse the output of the fast pass range reduction.
// |m_x| < 2^-7
double log10_accurate(int e_x, int index, double m_x) {
Float192 e_x_f192(static_cast<float>(e_x));
Float192 sum = fputil::quick_add(LOG10_R_F192[index],
fputil::quick_mul(LOG10_2, e_x_f192));
fputil::DoubleDouble mx{/*lo*/ 0.0, /*hi*/ m_x};
// Further range reductions.
double scale = 0x1.0p+7;
const fputil::DyadicFloat<128> NEG_ONE(-1.0);
for (size_t i = 0; i < R_STEPS; ++i) {
scale *= 0x1.0p+4;
int id = static_cast<int>(fputil::multiply_add(mx.hi, scale, 0x1.0p+4));
double r = RR[i][id];
fputil::DoubleDouble rm = fputil::exact_mult(r, mx.hi);
rm.hi += r - 1.0;
rm.lo = fputil::multiply_add(r, mx.lo, rm.lo);
mx = fputil::exact_add(rm.hi, rm.lo);
sum = fputil::quick_add(sum, LOG10_RR[i][id]);
}
// Now |m_x| <= 2^-27
Float192 m_hi(mx.hi);
Float192 m_lo(mx.lo);
Float192 m = fputil::quick_add(m_hi, m_lo);
Float192 p = fputil::quick_mul(m, BIG_COEFFS[0]);
for (size_t i = 1; i < 5; ++i) {
auto aa = fputil::quick_add(p, BIG_COEFFS[i]);
p = fputil::quick_mul(m, aa);
};
return static_cast<double>(fputil::quick_add(sum, p));
}
} // namespace
// TODO(lntue): Make the implementation correctly rounded for non-FMA targets.
LLVM_LIBC_FUNCTION(double, log10, (double x)) {
using FPBits_t = typename fputil::FPBits<double>;
FPBits_t xbits(x);
int x_e = -1023;
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;
}
// log10(x) = log10(2^x_e * x_m)
// = x_e * log10(2) + log10(x_m)
// Range reduction for log10(x_m):
// For each x_m, we would like to find R such that:
// |R * x_m - 1| < C
uint64_t x_u = xbits.uintval();
int shifted = x_u >> 45;
size_t index = shifted & 0x7F;
double r = R[index];
x_e += (x_u >> 52) & 0x7FF;
double e_x = static_cast<double>(x_e);
int e_err = (e_x == -1) && (index == 0x7F);
int logr_err = (index == 0);
double err =
fputil::multiply_add(e_x, LOG10_2_ULP[e_err], LOG10_R_ULP[logr_err]);
// hi is exact
double hi = fputil::multiply_add(e_x, LOG10_2_HI, LOG10_R[index].hi);
// lo errors ~ e_x * LSB(LOG10_2_LO) + LSB(LOG10_R[index].lo) + rounding err
// <= 2 * (e_x * LSB(LOG10_2_LO) + LSB(LOG10_R[index].lo))
double lo = fputil::multiply_add(e_x, LOG10_2_LO, LOG10_R[index].lo);
// A bound on the error is given
// in "Note on FastTwoSum with Directed Roundings"