diff options
Diffstat (limited to 'libc/src/math/generic/range_reduction_double_fma.h')
| -rw-r--r-- | libc/src/math/generic/range_reduction_double_fma.h | 495 |
1 files changed, 495 insertions, 0 deletions
diff --git a/libc/src/math/generic/range_reduction_double_fma.h b/libc/src/math/generic/range_reduction_double_fma.h new file mode 100644 index 0000000..c136de9 --- /dev/null +++ b/libc/src/math/generic/range_reduction_double_fma.h @@ -0,0 +1,495 @@ +//===-- Range reduction for double precision sin/cos/tan w/ FMA -*- 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_GENERIC_RANGE_REDUCTION_DOUBLE_FMA_H +#define LLVM_LIBC_SRC_MATH_GENERIC_RANGE_REDUCTION_DOUBLE_FMA_H + +#include "src/__support/FPUtil/FPBits.h" +#include "src/__support/FPUtil/double_double.h" +#include "src/__support/FPUtil/multiply_add.h" +#include "src/__support/FPUtil/nearest_integer.h" +#include "src/__support/common.h" + +namespace LIBC_NAMESPACE { + +namespace fma { + +using LIBC_NAMESPACE::fputil::DoubleDouble; + +LIBC_INLINE constexpr int FAST_PASS_EXPONENT = 32; + +// Digits of 2^(16*i) / pi, generated by Sollya with: +// For [2..62]: +// > for i from 3 to 63 do { +// pi_inv = 2^(16*(i - 3)) / pi; +// pn = nearestint(pi_inv); +// pi_frac = pi_inv - pn; +// a = round(pi_frac, D, RN); +// b = round(pi_frac - a, D, RN); +// c = round(pi_frac - a - b, D, RN); +// d = round(pi_frac - a - b - c, D, RN); +// print("{", 2^7 * a, ",", 2^7 * b, ",", 2^7 * c, ",", 2^7 * d, "},"); +// }; +// For [0..1]: +// The leading bit of 2^(16*(i - 3)) / pi is very small, so we add 0.25 so that +// the conditions for the algorithms are still satisfied, and one of those +// conditions guarantees that ulp(0.25 * x_reduced) >= 2, and will safely be +// discarded. +// for i from 0 to 2 do { +// pi_frac = 0.25 + 2^(16*(i - 3)) / pi; +// a = round(pi_frac, D, RN); +// b = round(pi_frac - a, D, RN); +// c = round(pi_frac - a - b, D, RN); +// d = round(pi_frac - a - b - c, D, RN); +// print("{", 2^7 * a, ",", 2^7 * b, ",", 2^7 * c, ",", 2^7 * d, "},"); +// }; +// For The fast pass using double-double, we only need 3 parts (a, b, c), but +// for the accurate pass using Float128, instead of using another table of +// Float128s, we simply add the fourth path (a, b, c, d), which simplify the +// implementation a bit and saving some memory. +LIBC_INLINE constexpr double ONE_TWENTY_EIGHT_OVER_PI[64][4] = { + {0x1.0000000000014p5, 0x1.7cc1b727220a9p-49, 0x1.3f84eafa3ea6ap-103, + -0x1.11f924eb53362p-157}, + {0x1.0000000145f3p5, 0x1.b727220a94fe1p-49, 0x1.d5f47d4d37703p-104, + 0x1.b6295993c439p-158}, + {0x1.000145f306dcap5, -0x1.bbead603d8a83p-50, 0x1.f534ddc0db629p-106, + 0x1.664f10e4107f9p-160}, + {0x1.45f306dc9c883p5, -0x1.6b01ec5417056p-49, -0x1.6447e493ad4cep-103, + 0x1.e21c820ff28b2p-157}, + {-0x1.f246c6efab581p4, 0x1.3abe8fa9a6eep-53, 0x1.b6c52b3278872p-107, + 0x1.07f9458eaf7afp-164}, + {0x1.391054a7f09d6p4, -0x1.70565911f924fp-53, 0x1.2b3278872084p-107, + -0x1.ae9c5421443aap-162}, + {0x1.529fc2757d1f5p2, 0x1.a6ee06db14acdp-53, -0x1.8778df7c035d4p-107, + 0x1.d5ef5de2b0db9p-161}, + {-0x1.ec54170565912p-1, 0x1.b6c52b3278872p-59, 0x1.07f9458eaf7afp-116, + -0x1.d4f246dc8e2dfp-173}, + {-0x1.505c1596447e5p5, 0x1.b14acc9e21c82p-49, 0x1.fe5163abdebbcp-106, + 0x1.586dc91b8e909p-160}, + {-0x1.596447e493ad5p1, 0x1.93c439041fe51p-54, 0x1.8eaf7aef1586ep-108, + -0x1.b7238b7b645a4p-163}, + {0x1.bb81b6c52b328p5, -0x1.de37df00d74e3p-49, 0x1.7bd778ac36e49p-103, + -0x1.1c5bdb22d1ffap-158}, + {0x1.b6c52b3278872p5, 0x1.07f9458eaf7afp-52, -0x1.d4f246dc8e2dfp-109, + 0x1.374b801924bbbp-164}, + {0x1.2b3278872084p5, -0x1.ae9c5421443aap-50, 0x1.b7246e3a424ddp-106, + 0x1.700324977504fp-161}, + {-0x1.8778df7c035d4p5, 0x1.d5ef5de2b0db9p-49, 0x1.1b8e909374b8p-104, + 0x1.924bba8274648p-160}, + {-0x1.bef806ba71508p4, -0x1.443a9e48db91cp-50, -0x1.6f6c8b47fe6dbp-104, + -0x1.115f62e6de302p-158}, + {-0x1.ae9c5421443aap-2, 0x1.b7246e3a424ddp-58, 0x1.700324977504fp-113, + -0x1.cdbc603c429c7p-167}, + {-0x1.38a84288753c9p5, -0x1.b7238b7b645a4p-51, 0x1.924bba8274648p-112, + 0x1.cfe1deb1cb12ap-166}, + {-0x1.0a21d4f246dc9p3, 0x1.d2126e9700325p-53, -0x1.a22bec5cdbc6p-107, + -0x1.e214e34ed658cp-162}, + {-0x1.d4f246dc8e2dfp3, 0x1.374b801924bbbp-52, -0x1.f62e6de301e21p-106, + -0x1.38d3b5963045ep-160}, + {-0x1.236e4716f6c8bp4, -0x1.1ff9b6d115f63p-50, 0x1.921cfe1deb1cbp-106, + 0x1.29a73ee88235fp-162}, + {0x1.b8e909374b802p4, -0x1.b6d115f62e6dep-50, -0x1.80f10a71a76b3p-105, + 0x1.cfba208d7d4bbp-160}, + {0x1.09374b801924cp4, -0x1.15f62e6de301ep-50, -0x1.0a71a76b2c609p-105, + 0x1.1046bea5d7689p-159}, + {-0x1.68ffcdb688afbp3, -0x1.736f180f10a72p-53, 0x1.62534e7dd1047p-107, + -0x1.0568a25dbd8b3p-161}, + {0x1.924bba8274648p0, 0x1.cfe1deb1cb12ap-54, -0x1.63045df7282b4p-108, + -0x1.44bb7b16638fep-162}, + {-0x1.a22bec5cdbc6p5, -0x1.e214e34ed658cp-50, -0x1.177dca0ad144cp-106, + 0x1.213a671c09ad1p-160}, + {0x1.3a32439fc3bd6p1, 0x1.cb129a73ee882p-54, 0x1.afa975da24275p-109, + -0x1.8e3f652e8207p-164}, + {-0x1.b78c0788538d4p4, 0x1.29a73ee88235fp-50, 0x1.4baed1213a672p-104, + -0x1.fb29741037d8dp-159}, + {0x1.fc3bd63962535p5, -0x1.822efb9415a29p-51, 0x1.a24274ce38136p-105, + -0x1.741037d8cdc54p-159}, + {-0x1.4e34ed658c117p2, -0x1.f7282b4512edfp-52, 0x1.d338e04d68bfp-107, + -0x1.bec66e29c67cbp-162}, + {0x1.62534e7dd1047p5, -0x1.0568a25dbd8b3p-49, -0x1.c7eca5d040df6p-105, + -0x1.9b8a719f2b318p-160}, + {-0x1.63045df7282b4p4, -0x1.44bb7b16638fep-50, 0x1.ad17df904e647p-104, + 0x1.639835339f49dp-158}, + {0x1.d1046bea5d769p5, -0x1.bd8b31c7eca5dp-49, -0x1.037d8cdc538dp-107, + 0x1.a99cfa4e422fcp-161}, + {0x1.afa975da24275p3, -0x1.8e3f652e8207p-52, 0x1.3991d63983534p-106, + -0x1.82d8dee81d108p-160}, + {-0x1.a28976f62cc72p5, 0x1.35a2fbf209cc9p-53, -0x1.4e33e566305b2p-109, + 0x1.08bf177bf2507p-163}, + {-0x1.76f62cc71fb29p5, -0x1.d040df633714ep-49, -0x1.9f2b3182d8defp-104, + 0x1.f8bbdf9283b2p-158}, + {0x1.d338e04d68bfp5, -0x1.bec66e29c67cbp-50, 0x1.9cfa4e422fc5ep-105, + -0x1.036be27003b4p-161}, + {0x1.c09ad17df904ep4, 0x1.91d639835339fp-50, 0x1.272117e2ef7e5p-104, + -0x1.7c4e007680022p-158}, + {0x1.68befc827323bp5, -0x1.c67cacc60b638p-50, 0x1.17e2ef7e4a0ecp-104, + 0x1.ff897ffde0598p-158}, + {-0x1.037d8cdc538dp5, 0x1.a99cfa4e422fcp-49, 0x1.77bf250763ff1p-103, + 0x1.7ffde05980fefp-158}, + {-0x1.8cdc538cf9599p5, 0x1.f49c845f8bbep-50, -0x1.b5f13801da001p-104, + 0x1.e05980fef2f12p-158}, + {-0x1.4e33e566305b2p3, 0x1.08bf177bf2507p-51, 0x1.8ffc4bffef02dp-105, + -0x1.fc04343b9d298p-160}, + {-0x1.f2b3182d8dee8p4, -0x1.d1081b5f13802p-52, 0x1.2fffbc0b301fep-107, + -0x1.a1dce94beb25cp-163}, + {-0x1.8c16c6f740e88p5, -0x1.036be27003b4p-49, -0x1.0fd33f8086877p-109, + -0x1.d297d64b824b2p-164}, + {0x1.3908bf177bf25p5, 0x1.d8ffc4bffef03p-53, -0x1.9fc04343b9d29p-108, + -0x1.f592e092c9813p-162}, + {0x1.7e2ef7e4a0ec8p4, -0x1.da00087e99fcp-56, -0x1.0d0ee74a5f593p-110, + 0x1.f6d367ecf27cbp-166}, + {-0x1.081b5f13801dap4, -0x1.0fd33f8086877p-61, -0x1.d297d64b824b2p-116, + -0x1.8130d834f648bp-170}, + {-0x1.af89c00ed0004p5, -0x1.fa67f010d0ee7p-50, -0x1.297d64b824b26p-104, + -0x1.30d834f648b0cp-162}, + {-0x1.c00ed00043f4dp5, 0x1.fde5e2316b415p-55, -0x1.2e092c98130d8p-110, + -0x1.a7b24585ce04dp-165}, + {0x1.2fffbc0b301fep5, -0x1.a1dce94beb25cp-51, -0x1.25930261b069fp-107, + 0x1.b74f463f669e6p-162}, + {-0x1.0fd33f8086877p3, -0x1.d297d64b824b2p-52, -0x1.8130d834f648bp-106, + -0x1.738132c3402bap-163}, + {-0x1.9fc04343b9d29p4, -0x1.f592e092c9813p-50, -0x1.b069ec9161738p-107, + -0x1.32c3402ba515bp-163}, + {-0x1.0d0ee74a5f593p2, 0x1.f6d367ecf27cbp-54, 0x1.36e9e8c7ecd3dp-111, + -0x1.00ae9456c229cp-165}, + {-0x1.dce94beb25c12p5, -0x1.64c0986c1a7b2p-49, -0x1.161738132c34p-103, + -0x1.5d28ad8453814p-158}, + {-0x1.4beb25c12593p5, -0x1.30d834f648b0cp-50, 0x1.8fd9a797fa8b6p-104, + -0x1.5b08a7028341dp-159}, + {0x1.b47db4d9fb3cap4, -0x1.a7b24585ce04dp-53, 0x1.3cbfd45aea4f7p-107, + 0x1.63f5f2f8bd9e8p-161}, + {-0x1.25930261b069fp5, 0x1.b74f463f669e6p-50, -0x1.5d28ad8453814p-110, + -0x1.a0e84c2f8c608p-166}, + {0x1.fb3c9f2c26dd4p4, -0x1.738132c3402bap-51, -0x1.456c229c0a0dp-105, + -0x1.d0985f18c10ebp-159}, + {-0x1.b069ec9161738p5, -0x1.32c3402ba515bp-51, -0x1.14e050683a131p-108, + 0x1.0739f78a5292fp-162}, + {-0x1.ec9161738132cp5, -0x1.a015d28ad8454p-50, 0x1.faf97c5ecf41dp-104, + -0x1.821d6b5b4565p-160}, + {-0x1.61738132c3403p5, 0x1.16ba93dd63f5fp-49, 0x1.7c5ecf41ce7dep-104, + 0x1.4a525d4d7f6bfp-159}, + {0x1.fb34f2ff516bbp3, -0x1.b08a7028341d1p-51, 0x1.9e839cfbc5295p-105, + -0x1.a2b2809409dc1p-159}, + {0x1.3cbfd45aea4f7p5, 0x1.63f5f2f8bd9e8p-49, 0x1.ce7de294a4baap-104, + -0x1.404a04ee072a3p-158}, + {-0x1.5d28ad8453814p2, -0x1.a0e84c2f8c608p-54, -0x1.d6b5b45650128p-108, + -0x1.3b81ca8bdea7fp-164}, + {-0x1.15b08a7028342p5, 0x1.7b3d0739f78a5p-50, 0x1.497535fdafd89p-105, + -0x1.ca8bdea7f33eep-164}, +}; + +// Lookup table for sin(k * pi / 128) with k = 0, ..., 255. +// Table is generated with Sollya as follow: +// > display = hexadecimal; +// > for k from 0 to 255 do { +// a = D(sin(k * pi/128)); }; +// b = D(sin(k * pi/128) - a); +// print("{", b, ",", a, "},"); +// }; +LIBC_INLINE constexpr DoubleDouble SIN_K_PI_OVER_128[256] = { + {0, 0}, + {-0x1.b1d63091a013p-64, 0x1.92155f7a3667ep-6}, + {-0x1.912bd0d569a9p-61, 0x1.91f65f10dd814p-5}, + {-0x1.9a088a8bf6b2cp-59, 0x1.2d52092ce19f6p-4}, + {-0x1.e2718d26ed688p-60, 0x1.917a6bc29b42cp-4}, + {0x1.a2704729ae56dp-59, 0x1.f564e56a9730ep-4}, + {0x1.13000a89a11ep-58, 0x1.2c8106e8e613ap-3}, + {0x1.531ff779ddac6p-57, 0x1.5e214448b3fc6p-3}, + {-0x1.26d19b9ff8d82p-57, 0x1.8f8b83c69a60bp-3}, + {-0x1.af1439e521935p-62, 0x1.c0b826a7e4f63p-3}, + {-0x1.42deef11da2c4p-57, 0x1.f19f97b215f1bp-3}, + {0x1.824c20ab7aa9ap-56, 0x1.111d262b1f677p-2}, + {-0x1.5d28da2c4612dp-56, 0x1.294062ed59f06p-2}, + {0x1.0c97c4afa2518p-56, 0x1.4135c94176601p-2}, + {-0x1.efdc0d58cf62p-62, 0x1.58f9a75ab1fddp-2}, + {-0x1.44b19e0864c5dp-56, 0x1.7088530fa459fp-2}, + {-0x1.72cedd3d5a61p-57, 0x1.87de2a6aea963p-2}, + {0x1.6da81290bdbabp-57, 0x1.9ef7943a8ed8ap-2}, + {0x1.5b362cb974183p-57, 0x1.b5d1009e15ccp-2}, + {0x1.6850e59c37f8fp-58, 0x1.cc66e9931c45ep-2}, + {0x1.e0d891d3c6841p-58, 0x1.e2b5d3806f63bp-2}, + {-0x1.2ec1fc1b776b8p-60, 0x1.f8ba4dbf89abap-2}, + {-0x1.a5a014347406cp-55, 0x1.073879922ffeep-1}, + {-0x1.ef23b69abe4f1p-55, 0x1.11eb3541b4b23p-1}, + {0x1.b25dd267f66p-55, 0x1.1c73b39ae68c8p-1}, + {-0x1.5da743ef3770cp-55, 0x1.26d054cdd12dfp-1}, + {-0x1.efcc626f74a6fp-57, 0x1.30ff7fce17035p-1}, + {0x1.e3e25e3954964p-56, 0x1.3affa292050b9p-1}, + {0x1.8076a2cfdc6b3p-57, 0x1.44cf325091dd6p-1}, + {0x1.3c293edceb327p-57, 0x1.4e6cabbe3e5e9p-1}, + {-0x1.75720992bfbb2p-55, 0x1.57d69348cecap-1}, + {-0x1.251b352ff2a37p-56, 0x1.610b7551d2cdfp-1}, + {-0x1.bdd3413b26456p-55, 0x1.6a09e667f3bcdp-1}, + {0x1.0d4ef0f1d915cp-55, 0x1.72d0837efff96p-1}, + {-0x1.0f537acdf0ad7p-56, 0x1.7b5df226aafafp-1}, + {-0x1.6f420f8ea3475p-56, 0x1.83b0e0bff976ep-1}, + {-0x1.2c5e12ed1336dp-55, 0x1.8bc806b151741p-1}, + {0x1.3d419a920df0bp-55, 0x1.93a22499263fbp-1}, + {-0x1.30ee286712474p-55, 0x1.9b3e047f38741p-1}, + {-0x1.128bb015df175p-56, 0x1.a29a7a0462782p-1}, + {0x1.9f630e8b6dac8p-60, 0x1.a9b66290ea1a3p-1}, + {-0x1.926da300ffccep-55, 0x1.b090a581502p-1}, + {-0x1.bc69f324e6d61p-55, 0x1.b728345196e3ep-1}, + {-0x1.825a732ac700ap-55, 0x1.bd7c0ac6f952ap-1}, + {-0x1.6e0b1757c8d07p-56, 0x1.c38b2f180bdb1p-1}, + {-0x1.2fb761e946603p-58, 0x1.c954b213411f5p-1}, + {-0x1.e7b6bb5ab58aep-58, 0x1.ced7af43cc773p-1}, + {-0x1.4ef5295d25af2p-55, 0x1.d4134d14dc93ap-1}, + {0x1.457e610231ac2p-56, 0x1.d906bcf328d46p-1}, + {0x1.83c37c6107db3p-55, 0x1.ddb13b6ccc23cp-1}, + {-0x1.014c76c126527p-55, 0x1.e212104f686e5p-1}, + {-0x1.16b56f2847754p-57, 0x1.e6288ec48e112p-1}, + {0x1.760b1e2e3f81ep-55, 0x1.e9f4156c62ddap-1}, + {0x1.e82c791f59cc2p-56, 0x1.ed740e7684963p-1}, + {0x1.52c7adc6b4989p-56, 0x1.f0a7efb9230d7p-1}, + {-0x1.d7bafb51f72e6p-56, 0x1.f38f3ac64e589p-1}, + {0x1.562172a361fd3p-56, 0x1.f6297cff75cbp-1}, + {0x1.ab256778ffcb6p-56, 0x1.f8764fa714ba9p-1}, + {-0x1.7a0a8ca13571fp-55, 0x1.fa7557f08a517p-1}, + {0x1.1ec8668ecaceep-55, 0x1.fc26470e19fd3p-1}, + {-0x1.87df6378811c7p-55, 0x1.fd88da3d12526p-1}, + {0x1.521ecd0c67e35p-57, 0x1.fe9cdad01883ap-1}, + {-0x1.c57bc2e24aa15p-57, 0x1.ff621e3796d7ep-1}, + {-0x1.1354d4556e4cbp-55, 0x1.ffd886084cd0dp-1}, + {0, 1}, + {-0x1.1354d4556e4cbp-55, 0x1.ffd886084cd0dp-1}, + {-0x1.c57bc2e24aa15p-57, 0x1.ff621e3796d7ep-1}, + {0x1.521ecd0c67e35p-57, 0x1.fe9cdad01883ap-1}, + {-0x1.87df6378811c7p-55, 0x1.fd88da3d12526p-1}, + {0x1.1ec8668ecaceep-55, 0x1.fc26470e19fd3p-1}, + {-0x1.7a0a8ca13571fp-55, 0x1.fa7557f08a517p-1}, + {0x1.ab256778ffcb6p-56, 0x1.f8764fa714ba9p-1}, + {0x1.562172a361fd3p-56, 0x1.f6297cff75cbp-1}, + {-0x1.d7bafb51f72e6p-56, 0x1.f38f3ac64e589p-1}, + {0x1.52c7adc6b4989p-56, 0x1.f0a7efb9230d7p-1}, + {0x1.e82c791f59cc2p-56, 0x1.ed740e7684963p-1}, + {0x1.760b1e2e3f81ep-55, 0x1.e9f4156c62ddap-1}, + {-0x1.16b56f2847754p-57, 0x1.e6288ec48e112p-1}, + {-0x1.014c76c126527p-55, 0x1.e212104f686e5p-1}, + {0x1.83c37c6107db3p-55, 0x1.ddb13b6ccc23cp-1}, + {0x1.457e610231ac2p-56, 0x1.d906bcf328d46p-1}, + {-0x1.4ef5295d25af2p-55, 0x1.d4134d14dc93ap-1}, + {-0x1.e7b6bb5ab58aep-58, 0x1.ced7af43cc773p-1}, + {-0x1.2fb761e946603p-58, 0x1.c954b213411f5p-1}, + {-0x1.6e0b1757c8d07p-56, 0x1.c38b2f180bdb1p-1}, + {-0x1.825a732ac700ap-55, 0x1.bd7c0ac6f952ap-1}, + {-0x1.bc69f324e6d61p-55, 0x1.b728345196e3ep-1}, + {-0x1.926da300ffccep-55, 0x1.b090a581502p-1}, + {0x1.9f630e8b6dac8p-60, 0x1.a9b66290ea1a3p-1}, + {-0x1.128bb015df175p-56, 0x1.a29a7a0462782p-1}, + {-0x1.30ee286712474p-55, 0x1.9b3e047f38741p-1}, + {0x1.3d419a920df0bp-55, 0x1.93a22499263fbp-1}, + {-0x1.2c5e12ed1336dp-55, 0x1.8bc806b151741p-1}, + {-0x1.6f420f8ea3475p-56, 0x1.83b0e0bff976ep-1}, + {-0x1.0f537acdf0ad7p-56, 0x1.7b5df226aafafp-1}, + {0x1.0d4ef0f1d915cp-55, 0x1.72d0837efff96p-1}, + {-0x1.bdd3413b26456p-55, 0x1.6a09e667f3bcdp-1}, + {-0x1.251b352ff2a37p-56, 0x1.610b7551d2cdfp-1}, + {-0x1.75720992bfbb2p-55, 0x1.57d69348cecap-1}, + {0x1.3c293edceb327p-57, 0x1.4e6cabbe3e5e9p-1}, + {0x1.8076a2cfdc6b3p-57, 0x1.44cf325091dd6p-1}, + {0x1.e3e25e3954964p-56, 0x1.3affa292050b9p-1}, + {-0x1.efcc626f74a6fp-57, 0x1.30ff7fce17035p-1}, + {-0x1.5da743ef3770cp-55, 0x1.26d054cdd12dfp-1}, + {0x1.b25dd267f66p-55, 0x1.1c73b39ae68c8p-1}, + {-0x1.ef23b69abe4f1p-55, 0x1.11eb3541b4b23p-1}, + {-0x1.a5a014347406cp-55, 0x1.073879922ffeep-1}, + {-0x1.2ec1fc1b776b8p-60, 0x1.f8ba4dbf89abap-2}, + {0x1.e0d891d3c6841p-58, 0x1.e2b5d3806f63bp-2}, + {0x1.6850e59c37f8fp-58, 0x1.cc66e9931c45ep-2}, + {0x1.5b362cb974183p-57, 0x1.b5d1009e15ccp-2}, + {0x1.6da81290bdbabp-57, 0x1.9ef7943a8ed8ap-2}, + {-0x1.72cedd3d5a61p-57, 0x1.87de2a6aea963p-2}, + {-0x1.44b19e0864c5dp-56, 0x1.7088530fa459fp-2}, + {-0x1.efdc0d58cf62p-62, 0x1.58f9a75ab1fddp-2}, + {0x1.0c97c4afa2518p-56, 0x1.4135c94176601p-2}, + {-0x1.5d28da2c4612dp-56, 0x1.294062ed59f06p-2}, + {0x1.824c20ab7aa9ap-56, 0x1.111d262b1f677p-2}, + {-0x1.42deef11da2c4p-57, 0x1.f19f97b215f1bp-3}, + {-0x1.af1439e521935p-62, 0x1.c0b826a7e4f63p-3}, + {-0x1.26d19b9ff8d82p-57, 0x1.8f8b83c69a60bp-3}, + {0x1.531ff779ddac6p-57, 0x1.5e214448b3fc6p-3}, + {0x1.13000a89a11ep-58, 0x1.2c8106e8e613ap-3}, + {0x1.a2704729ae56dp-59, 0x1.f564e56a9730ep-4}, + {-0x1.e2718d26ed688p-60, 0x1.917a6bc29b42cp-4}, + {-0x1.9a088a8bf6b2cp-59, 0x1.2d52092ce19f6p-4}, + {-0x1.912bd0d569a9p-61, 0x1.91f65f10dd814p-5}, + {-0x1.b1d63091a013p-64, 0x1.92155f7a3667ep-6}, + {0, 0}, + {0x1.b1d63091a013p-64, -0x1.92155f7a3667ep-6}, + {0x1.912bd0d569a9p-61, -0x1.91f65f10dd814p-5}, + {0x1.9a088a8bf6b2cp-59, -0x1.2d52092ce19f6p-4}, + {0x1.e2718d26ed688p-60, -0x1.917a6bc29b42cp-4}, + {-0x1.a2704729ae56dp-59, -0x1.f564e56a9730ep-4}, + {-0x1.13000a89a11ep-58, -0x1.2c8106e8e613ap-3}, + {-0x1.531ff779ddac6p-57, -0x1.5e214448b3fc6p-3}, + {0x1.26d19b9ff8d82p-57, -0x1.8f8b83c69a60bp-3}, + {0x1.af1439e521935p-62, -0x1.c0b826a7e4f63p-3}, + {0x1.42deef11da2c4p-57, -0x1.f19f97b215f1bp-3}, + {-0x1.824c20ab7aa9ap-56, -0x1.111d262b1f677p-2}, + {0x1.5d28da2c4612dp-56, -0x1.294062ed59f06p-2}, + {-0x1.0c97c4afa2518p-56, -0x1.4135c94176601p-2}, + {0x1.efdc0d58cf62p-62, -0x1.58f9a75ab1fddp-2}, + {0x1.44b19e0864c5dp-56, -0x1.7088530fa459fp-2}, + {0x1.72cedd3d5a61p-57, -0x1.87de2a6aea963p-2}, + {-0x1.6da81290bdbabp-57, -0x1.9ef7943a8ed8ap-2}, + {-0x1.5b362cb974183p-57, -0x1.b5d1009e15ccp-2}, + {-0x1.6850e59c37f8fp-58, -0x1.cc66e9931c45ep-2}, + {-0x1.e0d891d3c6841p-58, -0x1.e2b5d3806f63bp-2}, + {0x1.2ec1fc1b776b8p-60, -0x1.f8ba4dbf89abap-2}, + {0x1.a5a014347406cp-55, -0x1.073879922ffeep-1}, + {0x1.ef23b69abe4f1p-55, -0x1.11eb3541b4b23p-1}, + {-0x1.b25dd267f66p-55, -0x1.1c73b39ae68c8p-1}, + {0x1.5da743ef3770cp-55, -0x1.26d054cdd12dfp-1}, + {0x1.efcc626f74a6fp-57, -0x1.30ff7fce17035p-1}, + {-0x1.e3e25e3954964p-56, -0x1.3affa292050b9p-1}, + {-0x1.8076a2cfdc6b3p-57, -0x1.44cf325091dd6p-1}, + {-0x1.3c293edceb327p-57, -0x1.4e6cabbe3e5e9p-1}, + {0x1.75720992bfbb2p-55, -0x1.57d69348cecap-1}, + {0x1.251b352ff2a37p-56, -0x1.610b7551d2cdfp-1}, + {0x1.bdd3413b26456p-55, -0x1.6a09e667f3bcdp-1}, + {-0x1.0d4ef0f1d915cp-55, -0x1.72d0837efff96p-1}, + {0x1.0f537acdf0ad7p-56, -0x1.7b5df226aafafp-1}, + {0x1.6f420f8ea3475p-56, -0x1.83b0e0bff976ep-1}, + {0x1.2c5e12ed1336dp-55, -0x1.8bc806b151741p-1}, + {-0x1.3d419a920df0bp-55, -0x1.93a22499263fbp-1}, + {0x1.30ee286712474p-55, -0x1.9b3e047f38741p-1}, + {0x1.128bb015df175p-56, -0x1.a29a7a0462782p-1}, + {-0x1.9f630e8b6dac8p-60, -0x1.a9b66290ea1a3p-1}, + {0x1.926da300ffccep-55, -0x1.b090a581502p-1}, + {0x1.bc69f324e6d61p-55, -0x1.b728345196e3ep-1}, + {0x1.825a732ac700ap-55, -0x1.bd7c0ac6f952ap-1}, + {0x1.6e0b1757c8d07p-56, -0x1.c38b2f180bdb1p-1}, + {0x1.2fb761e946603p-58, -0x1.c954b213411f5p-1}, + {0x1.e7b6bb5ab58aep-58, -0x1.ced7af43cc773p-1}, + {0x1.4ef5295d25af2p-55, -0x1.d4134d14dc93ap-1}, + {-0x1.457e610231ac2p-56, -0x1.d906bcf328d46p-1}, + {-0x1.83c37c6107db3p-55, -0x1.ddb13b6ccc23cp-1}, + {0x1.014c76c126527p-55, -0x1.e212104f686e5p-1}, + {0x1.16b56f2847754p-57, -0x1.e6288ec48e112p-1}, + {-0x1.760b1e2e3f81ep-55, -0x1.e9f4156c62ddap-1}, + {-0x1.e82c791f59cc2p-56, -0x1.ed740e7684963p-1}, + {-0x1.52c7adc6b4989p-56, -0x1.f0a7efb9230d7p-1}, + {0x1.d7bafb51f72e6p-56, -0x1.f38f3ac64e589p-1}, + {-0x1.562172a361fd3p-56, -0x1.f6297cff75cbp-1}, + {-0x1.ab256778ffcb6p-56, -0x1.f8764fa714ba9p-1}, + {0x1.7a0a8ca13571fp-55, -0x1.fa7557f08a517p-1}, + {-0x1.1ec8668ecaceep-55, -0x1.fc26470e19fd3p-1}, + {0x1.87df6378811c7p-55, -0x1.fd88da3d12526p-1}, + {-0x1.521ecd0c67e35p-57, -0x1.fe9cdad01883ap-1}, + {0x1.c57bc2e24aa15p-57, -0x1.ff621e3796d7ep-1}, + {0x1.1354d4556e4cbp-55, -0x1.ffd886084cd0dp-1}, + {0, -1}, + {0x1.1354d4556e4cbp-55, -0x1.ffd886084cd0dp-1}, + {0x1.c57bc2e24aa15p-57, -0x1.ff621e3796d7ep-1}, + {-0x1.521ecd0c67e35p-57, -0x1.fe9cdad01883ap-1}, + {0x1.87df6378811c7p-55, -0x1.fd88da3d12526p-1}, + {-0x1.1ec8668ecaceep-55, -0x1.fc26470e19fd3p-1}, + {0x1.7a0a8ca13571fp-55, -0x1.fa7557f08a517p-1}, + {-0x1.ab256778ffcb6p-56, -0x1.f8764fa714ba9p-1}, + {-0x1.562172a361fd3p-56, -0x1.f6297cff75cbp-1}, + {0x1.d7bafb51f72e6p-56, -0x1.f38f3ac64e589p-1}, + {-0x1.52c7adc6b4989p-56, -0x1.f0a7efb9230d7p-1}, + {-0x1.e82c791f59cc2p-56, -0x1.ed740e7684963p-1}, + {-0x1.760b1e2e3f81ep-55, -0x1.e9f4156c62ddap-1}, + {0x1.16b56f2847754p-57, -0x1.e6288ec48e112p-1}, + {0x1.014c76c126527p-55, -0x1.e212104f686e5p-1}, + {-0x1.83c37c6107db3p-55, -0x1.ddb13b6ccc23cp-1}, + {-0x1.457e610231ac2p-56, -0x1.d906bcf328d46p-1}, + {0x1.4ef5295d25af2p-55, -0x1.d4134d14dc93ap-1}, + {0x1.e7b6bb5ab58aep-58, -0x1.ced7af43cc773p-1}, + {0x1.2fb761e946603p-58, -0x1.c954b213411f5p-1}, + {0x1.6e0b1757c8d07p-56, -0x1.c38b2f180bdb1p-1}, + {0x1.825a732ac700ap-55, -0x1.bd7c0ac6f952ap-1}, + {0x1.bc69f324e6d61p-55, -0x1.b728345196e3ep-1}, + {0x1.926da300ffccep-55, -0x1.b090a581502p-1}, + {-0x1.9f630e8b6dac8p-60, -0x1.a9b66290ea1a3p-1}, + {0x1.128bb015df175p-56, -0x1.a29a7a0462782p-1}, + {0x1.30ee286712474p-55, -0x1.9b3e047f38741p-1}, + {-0x1.3d419a920df0bp-55, -0x1.93a22499263fbp-1}, + {0x1.2c5e12ed1336dp-55, -0x1.8bc806b151741p-1}, + {0x1.6f420f8ea3475p-56, -0x1.83b0e0bff976ep-1}, + {0x1.0f537acdf0ad7p-56, -0x1.7b5df226aafafp-1}, + {-0x1.0d4ef0f1d915cp-55, -0x1.72d0837efff96p-1}, + {0x1.bdd3413b26456p-55, -0x1.6a09e667f3bcdp-1}, + {0x1.251b352ff2a37p-56, -0x1.610b7551d2cdfp-1}, + {0x1.75720992bfbb2p-55, -0x1.57d69348cecap-1}, + {-0x1.3c293edceb327p-57, -0x1.4e6cabbe3e5e9p-1}, + {-0x1.8076a2cfdc6b3p-57, -0x1.44cf325091dd6p-1}, + {-0x1.e3e25e3954964p-56, -0x1.3affa292050b9p-1}, + {0x1.efcc626f74a6fp-57, -0x1.30ff7fce17035p-1}, + {0x1.5da743ef3770cp-55, -0x1.26d054cdd12dfp-1}, + {-0x1.b25dd267f66p-55, -0x1.1c73b39ae68c8p-1}, + {0x1.ef23b69abe4f1p-55, -0x1.11eb3541b4b23p-1}, + {0x1.a5a014347406cp-55, -0x1.073879922ffeep-1}, + {0x1.2ec1fc1b776b8p-60, -0x1.f8ba4dbf89abap-2}, + {-0x1.e0d891d3c6841p-58, -0x1.e2b5d3806f63bp-2}, + {-0x1.6850e59c37f8fp-58, -0x1.cc66e9931c45ep-2}, + {-0x1.5b362cb974183p-57, -0x1.b5d1009e15ccp-2}, + {-0x1.6da81290bdbabp-57, -0x1.9ef7943a8ed8ap-2}, + {0x1.72cedd3d5a61p-57, -0x1.87de2a6aea963p-2}, + {0x1.44b19e0864c5dp-56, -0x1.7088530fa459fp-2}, + {0x1.efdc0d58cf62p-62, -0x1.58f9a75ab1fddp-2}, + {-0x1.0c97c4afa2518p-56, -0x1.4135c94176601p-2}, + {0x1.5d28da2c4612dp-56, -0x1.294062ed59f06p-2}, + {-0x1.824c20ab7aa9ap-56, -0x1.111d262b1f677p-2}, + {0x1.42deef11da2c4p-57, -0x1.f19f97b215f1bp-3}, + {0x1.af1439e521935p-62, -0x1.c0b826a7e4f63p-3}, + {0x1.26d19b9ff8d82p-57, -0x1.8f8b83c69a60bp-3}, + {-0x1.531ff779ddac6p-57, -0x1.5e214448b3fc6p-3}, + {-0x1.13000a89a11ep-58, -0x1.2c8106e8e613ap-3}, + {-0x1.a2704729ae56dp-59, -0x1.f564e56a9730ep-4}, + {0x1.e2718d26ed688p-60, -0x1.917a6bc29b42cp-4}, + {0x1.9a088a8bf6b2cp-59, -0x1.2d52092ce19f6p-4}, + {0x1.912bd0d569a9p-61, -0x1.91f65f10dd814p-5}, + {0x1.b1d63091a013p-64, -0x1.92155f7a3667ep-6}, +}; + +// For |x| < 2^-32, return k and u such that: +// k = round(x * 128/pi) +// x mod pi/128 = x - k * pi/128 ~ u.hi + u.lo +LIBC_INLINE unsigned range_reduction_small(double x, DoubleDouble &u) { + // Digits of pi/128, generated by Sollya with: + // > a = round(pi/128, D, RN); + // > b = round(pi/128 - a, D, RN); + constexpr DoubleDouble PI_OVER_128_DD = {0x1.1a62633145c07p-60, + 0x1.921fb54442d18p-6}; + + double prod_hi = x * ONE_TWENTY_EIGHT_OVER_PI[3][0]; + double kd = fputil::nearest_integer(prod_hi); + + // Let y = x - k * (pi/128) + // Then |y| < pi / 256 + // With extra rounding errors, we can bound |y| < 2^-6. + double y_hi = fputil::multiply_add(kd, -PI_OVER_128_DD.hi, x); // Exact + // u_hi + u_lo ~ (y_hi + kd*(-PI_OVER_128_DD[1])) + // and |u_lo| < 2* ulp(u_hi) + // The upper bound 2^-6 is over-estimated, we should still have: + // |u_hi + u_lo| < 2^-6. + u.hi = fputil::multiply_add(kd, -PI_OVER_128_DD.lo, y_hi); + u.lo = y_hi - u.hi; // Exact; + u.lo = fputil::multiply_add(kd, -PI_OVER_128_DD.lo, u.lo); + // Error bound: + // For |x| < 2^32: + // |x * high part of 128/pi| < 2^32 * 2^6 = 2^38 + // So |k| = |round(x * high part of 128/pi)| < 2^38 + // And hence, + // |(x mod pi/128) - (u.hi + u.lo)| <= ulp(2 * kd * PI_OVER_128_DD.lo) + // < 2 * 2^38 * 2^-59 * 2^-52 + // = 2^-72 + // Note: if we limit the input exponent to the same as in non-FMA version, + // i.e., |x| < 2^-23, then the output errors can be bounded by 2^-81, similar + // to the large range reduction bound. + return static_cast<unsigned>(static_cast<int64_t>(kd)); +} + +} // namespace fma + +} // namespace LIBC_NAMESPACE + +#endif // LLVM_LIBC_SRC_MATH_GENERIC_RANGE_REDUCTION_DOUBLE_FMA_H |