diff options
Diffstat (limited to 'libc/src/math/generic/range_reduction_double_nofma.h')
| -rw-r--r-- | libc/src/math/generic/range_reduction_double_nofma.h | 493 |
1 files changed, 493 insertions, 0 deletions
diff --git a/libc/src/math/generic/range_reduction_double_nofma.h b/libc/src/math/generic/range_reduction_double_nofma.h new file mode 100644 index 0000000..b9d34d6 --- /dev/null +++ b/libc/src/math/generic/range_reduction_double_nofma.h @@ -0,0 +1,493 @@ +//===-- Range reduction for double precision sin/cos/tan --------*- 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_NOFMA_H +#define LLVM_LIBC_SRC_MATH_GENERIC_RANGE_REDUCTION_DOUBLE_NOFMA_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 nofma { + +using fputil::DoubleDouble; + +LIBC_INLINE constexpr int FAST_PASS_EXPONENT = 23; + +// 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, 51, RN); +// b = round(pi_frac - a, 51, 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, 51, RN); +// b = round(pi_frac - a, 51, 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, 0x1.7cc1b727220a8p-49, 0x1.4fe13abe8fa9ap-101, + 0x1.bb81b6c52b328p-155}, + {0x1.0000000145f3p5, 0x1.b727220a94fep-49, 0x1.3abe8fa9a6eep-101, + 0x1.b6c52b3278872p-155}, + {0x1.000145f306dc8p5, 0x1.c882a53f84ebp-47, -0x1.70565911f924fp-101, + 0x1.2b3278872084p-155}, + {0x1.45f306dc9c884p5, -0x1.5ac07b1505c14p-47, -0x1.96447e493ad4dp-99, + 0x1.3c439041fe516p-154}, + {-0x1.f246c6efab58p4, -0x1.ec5417056591p-49, -0x1.f924eb53361dep-101, + -0x1.bef806ba71508p-156}, + {0x1.391054a7f09d4p4, 0x1.f47d4d377036cp-48, 0x1.8a5664f10e41p-100, + 0x1.fe5163abdebbcp-154}, + {0x1.529fc2757d1f4p2, 0x1.34ddc0db62958p-50, 0x1.93c439041fe51p-102, + 0x1.8eaf7aef1586ep-156}, + {-0x1.ec5417056591p-1, -0x1.f924eb53361ep-53, 0x1.c820ff28b1d5fp-105, + -0x1.443a9e48db91cp-162}, + {-0x1.505c1596447e4p5, -0x1.275a99b0ef1cp-48, 0x1.07f9458eaf7afp-100, + -0x1.d4f246dc8e2dfp-157}, + {-0x1.596447e493ad4p1, -0x1.9b0ef1bef806cp-52, 0x1.63abdebbc561bp-106, + 0x1.c91b8e909374cp-160}, + {0x1.bb81b6c52b328p5, -0x1.de37df00d74e4p-49, 0x1.5ef5de2b0db92p-101, + 0x1.b8e909374b802p-156}, + {0x1.b6c52b3278874p5, -0x1.f7c035d38a844p-47, 0x1.778ac36e48dc7p-99, + 0x1.2126e97003249p-153}, + {0x1.2b3278872084p5, -0x1.ae9c5421443a8p-50, -0x1.e48db91c5bdb2p-102, + -0x1.68ffcdb688afbp-157}, + {-0x1.8778df7c035d4p5, 0x1.d5ef5de2b0db8p-49, 0x1.2371d2126e97p-101, + 0x1.924bba8274648p-160}, + {-0x1.bef806ba71508p4, -0x1.443a9e48db91cp-50, -0x1.6f6c8b47fe6dbp-104, + -0x1.115f62e6de302p-158}, + {-0x1.ae9c5421443a8p-2, -0x1.e48db91c5bdb4p-54, 0x1.d2e006492eea1p-106, + -0x1.8b9b78c078854p-160}, + {-0x1.38a84288753c8p5, -0x1.1b7238b7b645cp-47, 0x1.c00c925dd413ap-99, + 0x1.921cfe1deb1cbp-154}, + {-0x1.0a21d4f246dc8p3, -0x1.c5bdb22d1ff9cp-50, 0x1.25dd413a3243ap-103, + -0x1.e214e34ed658cp-162}, + {-0x1.d4f246dc8e2ep3, 0x1.26e9700324978p-49, -0x1.5f62e6de301e2p-102, + -0x1.4e34ed658c117p-158}, + {-0x1.236e4716f6c8cp4, 0x1.700324977505p-49, -0x1.736f180f10a72p-101, + 0x1.62534e7dd1047p-155}, + {0x1.b8e909374b8p4, 0x1.924bba8274648p-48, 0x1.cfe1deb1cb12ap-102, + -0x1.63045df7282b4p-156}, + {0x1.09374b801924cp4, -0x1.15f62e6de302p-50, 0x1.deb1cb129a73fp-102, + -0x1.77dca0ad144bbp-158}, + {-0x1.68ffcdb688afcp3, 0x1.d1921cfe1debp-50, 0x1.cb129a73ee882p-102, + 0x1.afa975da24275p-157}, + {0x1.924bba8274648p0, 0x1.cfe1deb1cb128p-54, 0x1.a73ee88235f53p-106, + -0x1.44bb7b16638fep-162}, + {-0x1.a22bec5cdbc6p5, -0x1.e214e34ed658cp-50, -0x1.177dca0ad144cp-106, + 0x1.213a671c09ad1p-160}, + {0x1.3a32439fc3bd8p1, -0x1.c69dacb1822fp-51, 0x1.1afa975da2427p-105, + 0x1.338e04d68befdp-159}, + {-0x1.b78c0788538d4p4, 0x1.29a73ee88236p-50, -0x1.5a28976f62cc7p-103, + -0x1.fb29741037d8dp-159}, + {0x1.fc3bd63962534p5, 0x1.cfba208d7d4bcp-48, -0x1.12edec598e3f6p-100, + -0x1.4ba081bec66e3p-154}, + {-0x1.4e34ed658c118p2, 0x1.046bea5d7689p-51, 0x1.3a671c09ad17ep-104, + -0x1.bec66e29c67cbp-162}, + {0x1.62534e7dd1048p5, -0x1.415a28976f62cp-47, -0x1.8e3f652e8207p-100, + 0x1.3991d63983534p-154}, + {-0x1.63045df7282b4p4, -0x1.44bb7b16638fcp-50, -0x1.94ba081bec66ep-102, + -0x1.4e33e566305b2p-157}, + {0x1.d1046bea5d768p5, 0x1.213a671c09adp-48, 0x1.7df904e64758ep-100, + 0x1.835339f49c846p-154}, + {0x1.afa975da24274p3, 0x1.9c7026b45f7e4p-50, 0x1.3991d63983534p-106, + -0x1.82d8dee81d108p-160}, + {-0x1.a28976f62cc7p5, -0x1.fb29741037d8cp-47, -0x1.b8a719f2b3183p-100, + 0x1.3908bf177bf25p-155}, + {-0x1.76f62cc71fb28p5, -0x1.741037d8cdc54p-47, 0x1.cc1a99cfa4e42p-101, + 0x1.7e2ef7e4a0ec8p-156}, + {0x1.d338e04d68bfp5, -0x1.bec66e29c67ccp-50, 0x1.339f49c845f8cp-102, + -0x1.081b5f13801dap-156}, + {0x1.c09ad17df905p4, -0x1.9b8a719f2b318p-48, -0x1.6c6f740e8840ep-103, + 0x1.41d8ffc4bffefp-157}, + {0x1.68befc827323cp5, -0x1.38cf9598c16c8p-47, 0x1.08bf177bf2507p-99, + 0x1.8ffc4bffef02dp-153}, + {-0x1.037d8cdc538dp5, 0x1.a99cfa4e422fcp-49, 0x1.77bf250763ff1p-103, + 0x1.7ffde05980fefp-158}, + {-0x1.8cdc538cf9598p5, -0x1.82d8dee81d108p-48, -0x1.b5f13801da001p-104, + 0x1.e05980fef2f12p-158}, + {-0x1.4e33e566305bp3, -0x1.bdd03a21036cp-49, 0x1.d8ffc4bffef03p-101, + -0x1.9fc04343b9d29p-156}, + {-0x1.f2b3182d8dee8p4, -0x1.d1081b5f138p-52, -0x1.da00087e99fcp-104, + -0x1.0d0ee74a5f593p-158}, + {-0x1.8c16c6f740e88p5, -0x1.036be27003b4p-49, -0x1.0fd33f8086877p-109, + -0x1.d297d64b824b2p-164}, + {0x1.3908bf177bf24p5, 0x1.0763ff12fffbcp-47, 0x1.6603fbcbc462dp-104, + 0x1.a0a6d1f6d367fp-158}, + {0x1.7e2ef7e4a0ec8p4, -0x1.da00087e99fcp-56, -0x1.0d0ee74a5f593p-110, + 0x1.f6d367ecf27cbp-166}, + {-0x1.081b5f13801dcp4, 0x1.fff7816603fbcp-48, 0x1.788c5ad05369p-101, + -0x1.25930261b069fp-155}, + {-0x1.af89c00ed0004p5, -0x1.fa67f010d0ee8p-50, 0x1.6b414da3eda6dp-103, + -0x1.30d834f648b0cp-162}, + {-0x1.c00ed00043f4cp5, -0x1.fc04343b9d298p-48, 0x1.4da3eda6cfd9ep-103, + 0x1.3e584dba7a32p-157}, + {0x1.2fffbc0b301fcp5, 0x1.e5e2316b414dcp-47, -0x1.c125930261b07p-99, + 0x1.84dba7a31fb35p-153}, + {-0x1.0fd33f8086878p3, 0x1.8b5a0a6d1f6d4p-50, -0x1.30261b069ec91p-103, + -0x1.85ce04cb0d00bp-157}, + {-0x1.9fc04343b9d28p4, -0x1.7d64b824b2604p-48, -0x1.86c1a7b24585dp-101, + 0x1.fb34f2ff516bbp-157}, + {-0x1.0d0ee74a5f594p2, 0x1.1f6d367ecf27cp-50, 0x1.6136e9e8c7ecdp-103, + 0x1.e5fea2d7527bbp-158}, + {-0x1.dce94beb25c14p5, 0x1.a6cfd9e4f9614p-47, -0x1.22c2e70265868p-100, + -0x1.5d28ad8453814p-158}, + {-0x1.4beb25c12593p5, -0x1.30d834f648b0cp-50, 0x1.8fd9a797fa8b6p-104, + -0x1.5b08a7028341dp-159}, + {0x1.b47db4d9fb3c8p4, 0x1.f2c26dd3d18fcp-48, 0x1.9a797fa8b5d4ap-100, + -0x1.14e050683a131p-156}, + {-0x1.25930261b06ap5, 0x1.36e9e8c7ecd3cp-47, 0x1.7fa8b5d49eeb2p-100, + -0x1.41a0e84c2f8c6p-158}, + {0x1.fb3c9f2c26dd4p4, -0x1.738132c3402bcp-51, 0x1.aea4f758fd7ccp-103, + -0x1.d0985f18c10ebp-159}, + {-0x1.b069ec9161738p5, -0x1.32c3402ba515cp-51, 0x1.eeb1faf97c5edp-104, + -0x1.7c63043ad6b69p-161}, + {-0x1.ec9161738132cp5, -0x1.a015d28ad8454p-50, 0x1.faf97c5ecf41dp-104, + -0x1.821d6b5b4565p-160}, + {-0x1.61738132c3404p5, 0x1.45aea4f758fd8p-47, -0x1.a0e84c2f8c608p-102, + -0x1.d6b5b45650128p-156}, + {0x1.fb34f2ff516bcp3, -0x1.6c229c0a0d074p-49, -0x1.30be31821d6b6p-104, + 0x1.2ea6bfb5fb12p-158}, + {0x1.3cbfd45aea4f8p5, -0x1.4e050683a130cp-48, 0x1.ce7de294a4baap-104, + -0x1.404a04ee072a3p-158}, + {-0x1.5d28ad8453814p2, -0x1.a0e84c2f8c608p-54, -0x1.d6b5b45650128p-108, + -0x1.3b81ca8bdea7fp-164}, + {-0x1.15b08a702834p5, -0x1.d0985f18c10ecp-47, 0x1.4a4ba9afed7ecp-100, + 0x1.1f8d5d0856033p-154}, +}; + +// 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 = round(sin(k * pi/128), 51, RN); +// b = round(sin(k * pi/128) - a, D, RN); +// print("{", b, ",", a, "},"); +// }; +LIBC_INLINE constexpr DoubleDouble SIN_K_PI_OVER_128[256] = { + {0, 0}, + {0x1.f938a73db97fbp-58, 0x1.92155f7a3667cp-6}, + {-0x1.912bd0d569a9p-61, 0x1.91f65f10dd814p-5}, + {0x1.ccbeeeae8129ap-56, 0x1.2d52092ce19f4p-4}, + {-0x1.e2718d26ed688p-60, 0x1.917a6bc29b42cp-4}, + {-0x1.cbb1f71aca352p-56, 0x1.f564e56a9731p-4}, + {-0x1.dd9ffeaecbdc4p-55, 0x1.2c8106e8e613cp-3}, + {-0x1.ab3802218894fp-55, 0x1.5e214448b3fc8p-3}, + {-0x1.49b466e7fe36p-55, 0x1.8f8b83c69a60cp-3}, + {-0x1.035e2873ca432p-55, 0x1.c0b826a7e4f64p-3}, + {-0x1.50b7bbc4768b1p-55, 0x1.f19f97b215f1cp-3}, + {-0x1.3ed9efaa42ab3p-55, 0x1.111d262b1f678p-2}, + {0x1.a8b5c974ee7b5p-54, 0x1.294062ed59f04p-2}, + {0x1.4325f12be8946p-54, 0x1.4135c941766p-2}, + {0x1.fc2047e54e614p-55, 0x1.58f9a75ab1fdcp-2}, + {-0x1.512c678219317p-54, 0x1.7088530fa45ap-2}, + {-0x1.2e59dba7ab4c2p-54, 0x1.87de2a6aea964p-2}, + {-0x1.d24afdade848bp-54, 0x1.9ef7943a8ed8cp-2}, + {0x1.5b362cb974183p-57, 0x1.b5d1009e15ccp-2}, + {-0x1.e97af1a63c807p-54, 0x1.cc66e9931c46p-2}, + {-0x1.c3e4edc5872f8p-55, 0x1.e2b5d3806f63cp-2}, + {0x1.fb44f80f92225p-54, 0x1.f8ba4dbf89ab8p-2}, + {0x1.9697faf2e2fe5p-53, 0x1.073879922ffecp-1}, + {-0x1.7bc8eda6af93cp-53, 0x1.11eb3541b4b24p-1}, + {0x1.b25dd267f66p-55, 0x1.1c73b39ae68c8p-1}, + {-0x1.5769d0fbcddc3p-53, 0x1.26d054cdd12ep-1}, + {0x1.c20673b2116b2p-54, 0x1.30ff7fce17034p-1}, + {0x1.3c7c4bc72a92cp-53, 0x1.3affa292050b8p-1}, + {-0x1.e7f895d302395p-53, 0x1.44cf325091dd8p-1}, + {0x1.13c293edceb32p-53, 0x1.4e6cabbe3e5e8p-1}, + {-0x1.75720992bfbb2p-55, 0x1.57d69348cecap-1}, + {-0x1.24a366a5fe547p-53, 0x1.610b7551d2cep-1}, + {0x1.21165f626cdd5p-54, 0x1.6a09e667f3bccp-1}, + {-0x1.bcac43c389ba9p-53, 0x1.72d0837efff98p-1}, + {-0x1.21ea6f59be15bp-53, 0x1.7b5df226aafbp-1}, + {0x1.d217be0e2b971p-53, 0x1.83b0e0bff976cp-1}, + {0x1.69d0f6897664ap-54, 0x1.8bc806b15174p-1}, + {-0x1.615f32b6f907ap-54, 0x1.93a22499263fcp-1}, + {0x1.6788ebcc76dc6p-54, 0x1.9b3e047f3874p-1}, + {0x1.ddae89fd441d1p-53, 0x1.a29a7a046278p-1}, + {-0x1.f98273c5d2495p-54, 0x1.a9b66290ea1a4p-1}, + {-0x1.926da300ffccep-55, 0x1.b090a581502p-1}, + {0x1.90e58336c64a8p-53, 0x1.b728345196e3cp-1}, + {0x1.9f6963354e3fep-53, 0x1.bd7c0ac6f9528p-1}, + {0x1.a47d3a2a0dcbep-54, 0x1.c38b2f180bdbp-1}, + {0x1.ed0489e16b9ap-54, 0x1.c954b213411f4p-1}, + {-0x1.0f3db5dad5ac5p-53, 0x1.ced7af43cc774p-1}, + {0x1.ac42b5a8b6943p-53, 0x1.d4134d14dc938p-1}, + {-0x1.d75033dfb9ca8p-53, 0x1.d906bcf328d48p-1}, + {0x1.83c37c6107db3p-55, 0x1.ddb13b6ccc23cp-1}, + {0x1.7f59c49f6cd6dp-54, 0x1.e212104f686e4p-1}, + {0x1.ee94a90d7b88bp-53, 0x1.e6288ec48e11p-1}, + {-0x1.a27d3874701f9p-53, 0x1.e9f4156c62ddcp-1}, + {-0x1.85f4e1b8298dp-54, 0x1.ed740e7684964p-1}, + {-0x1.ab4e148e52d9ep-54, 0x1.f0a7efb9230d8p-1}, + {0x1.8a11412b82346p-54, 0x1.f38f3ac64e588p-1}, + {0x1.562172a361fd3p-56, 0x1.f6297cff75cbp-1}, + {0x1.3564acef1ff97p-53, 0x1.f8764fa714ba8p-1}, + {-0x1.5e82a3284d5c8p-53, 0x1.fa7557f08a518p-1}, + {-0x1.709bccb89a989p-54, 0x1.fc26470e19fd4p-1}, + {0x1.9e082721dfb8ep-53, 0x1.fd88da3d12524p-1}, + {-0x1.eade132f3981dp-53, 0x1.fe9cdad01883cp-1}, + {0x1.e3a843d1db55fp-53, 0x1.ff621e3796d7cp-1}, + {0x1.765595d548d9ap-54, 0x1.ffd886084cd0cp-1}, + {0, 1}, + {0x1.765595d548d9ap-54, 0x1.ffd886084cd0cp-1}, + {0x1.e3a843d1db55fp-53, 0x1.ff621e3796d7cp-1}, + {-0x1.eade132f3981dp-53, 0x1.fe9cdad01883cp-1}, + {0x1.9e082721dfb8ep-53, 0x1.fd88da3d12524p-1}, + {-0x1.709bccb89a989p-54, 0x1.fc26470e19fd4p-1}, + {-0x1.5e82a3284d5c8p-53, 0x1.fa7557f08a518p-1}, + {0x1.3564acef1ff97p-53, 0x1.f8764fa714ba8p-1}, + {0x1.562172a361fd3p-56, 0x1.f6297cff75cbp-1}, + {0x1.8a11412b82346p-54, 0x1.f38f3ac64e588p-1}, + {-0x1.ab4e148e52d9ep-54, 0x1.f0a7efb9230d8p-1}, + {-0x1.85f4e1b8298dp-54, 0x1.ed740e7684964p-1}, + {-0x1.a27d3874701f9p-53, 0x1.e9f4156c62ddcp-1}, + {0x1.ee94a90d7b88bp-53, 0x1.e6288ec48e11p-1}, + {0x1.7f59c49f6cd6dp-54, 0x1.e212104f686e4p-1}, + {0x1.83c37c6107db3p-55, 0x1.ddb13b6ccc23cp-1}, + {-0x1.d75033dfb9ca8p-53, 0x1.d906bcf328d48p-1}, + {0x1.ac42b5a8b6943p-53, 0x1.d4134d14dc938p-1}, + {-0x1.0f3db5dad5ac5p-53, 0x1.ced7af43cc774p-1}, + {0x1.ed0489e16b9ap-54, 0x1.c954b213411f4p-1}, + {0x1.a47d3a2a0dcbep-54, 0x1.c38b2f180bdbp-1}, + {0x1.9f6963354e3fep-53, 0x1.bd7c0ac6f9528p-1}, + {0x1.90e58336c64a8p-53, 0x1.b728345196e3cp-1}, + {-0x1.926da300ffccep-55, 0x1.b090a581502p-1}, + {-0x1.f98273c5d2495p-54, 0x1.a9b66290ea1a4p-1}, + {0x1.ddae89fd441d1p-53, 0x1.a29a7a046278p-1}, + {0x1.6788ebcc76dc6p-54, 0x1.9b3e047f3874p-1}, + {-0x1.615f32b6f907ap-54, 0x1.93a22499263fcp-1}, + {0x1.69d0f6897664ap-54, 0x1.8bc806b15174p-1}, + {0x1.d217be0e2b971p-53, 0x1.83b0e0bff976cp-1}, + {-0x1.21ea6f59be15bp-53, 0x1.7b5df226aafbp-1}, + {-0x1.bcac43c389ba9p-53, 0x1.72d0837efff98p-1}, + {0x1.21165f626cdd5p-54, 0x1.6a09e667f3bccp-1}, + {-0x1.24a366a5fe547p-53, 0x1.610b7551d2cep-1}, + {-0x1.75720992bfbb2p-55, 0x1.57d69348cecap-1}, + {0x1.13c293edceb32p-53, 0x1.4e6cabbe3e5e8p-1}, + {-0x1.e7f895d302395p-53, 0x1.44cf325091dd8p-1}, + {0x1.3c7c4bc72a92cp-53, 0x1.3affa292050b8p-1}, + {0x1.c20673b2116b2p-54, 0x1.30ff7fce17034p-1}, + {-0x1.5769d0fbcddc3p-53, 0x1.26d054cdd12ep-1}, + {0x1.b25dd267f66p-55, 0x1.1c73b39ae68c8p-1}, + {-0x1.7bc8eda6af93cp-53, 0x1.11eb3541b4b24p-1}, + {0x1.9697faf2e2fe5p-53, 0x1.073879922ffecp-1}, + {0x1.fb44f80f92225p-54, 0x1.f8ba4dbf89ab8p-2}, + {-0x1.c3e4edc5872f8p-55, 0x1.e2b5d3806f63cp-2}, + {-0x1.e97af1a63c807p-54, 0x1.cc66e9931c46p-2}, + {0x1.5b362cb974183p-57, 0x1.b5d1009e15ccp-2}, + {-0x1.d24afdade848bp-54, 0x1.9ef7943a8ed8cp-2}, + {-0x1.2e59dba7ab4c2p-54, 0x1.87de2a6aea964p-2}, + {-0x1.512c678219317p-54, 0x1.7088530fa45ap-2}, + {0x1.fc2047e54e614p-55, 0x1.58f9a75ab1fdcp-2}, + {0x1.4325f12be8946p-54, 0x1.4135c941766p-2}, + {0x1.a8b5c974ee7b5p-54, 0x1.294062ed59f04p-2}, + {-0x1.3ed9efaa42ab3p-55, 0x1.111d262b1f678p-2}, + {-0x1.50b7bbc4768b1p-55, 0x1.f19f97b215f1cp-3}, + {-0x1.035e2873ca432p-55, 0x1.c0b826a7e4f64p-3}, + {-0x1.49b466e7fe36p-55, 0x1.8f8b83c69a60cp-3}, + {-0x1.ab3802218894fp-55, 0x1.5e214448b3fc8p-3}, + {-0x1.dd9ffeaecbdc4p-55, 0x1.2c8106e8e613cp-3}, + {-0x1.cbb1f71aca352p-56, 0x1.f564e56a9731p-4}, + {-0x1.e2718d26ed688p-60, 0x1.917a6bc29b42cp-4}, + {0x1.ccbeeeae8129ap-56, 0x1.2d52092ce19f4p-4}, + {-0x1.912bd0d569a9p-61, 0x1.91f65f10dd814p-5}, + {0x1.f938a73db97fbp-58, 0x1.92155f7a3667cp-6}, + {0, 0}, + {-0x1.f938a73db97fbp-58, -0x1.92155f7a3667cp-6}, + {0x1.912bd0d569a9p-61, -0x1.91f65f10dd814p-5}, + {-0x1.ccbeeeae8129ap-56, -0x1.2d52092ce19f4p-4}, + {0x1.e2718d26ed688p-60, -0x1.917a6bc29b42cp-4}, + {0x1.cbb1f71aca352p-56, -0x1.f564e56a9731p-4}, + {0x1.dd9ffeaecbdc4p-55, -0x1.2c8106e8e613cp-3}, + {0x1.ab3802218894fp-55, -0x1.5e214448b3fc8p-3}, + {0x1.49b466e7fe36p-55, -0x1.8f8b83c69a60cp-3}, + {0x1.035e2873ca432p-55, -0x1.c0b826a7e4f64p-3}, + {0x1.50b7bbc4768b1p-55, -0x1.f19f97b215f1cp-3}, + {0x1.3ed9efaa42ab3p-55, -0x1.111d262b1f678p-2}, + {-0x1.a8b5c974ee7b5p-54, -0x1.294062ed59f04p-2}, + {-0x1.4325f12be8946p-54, -0x1.4135c941766p-2}, + {-0x1.fc2047e54e614p-55, -0x1.58f9a75ab1fdcp-2}, + {0x1.512c678219317p-54, -0x1.7088530fa45ap-2}, + {0x1.2e59dba7ab4c2p-54, -0x1.87de2a6aea964p-2}, + {0x1.d24afdade848bp-54, -0x1.9ef7943a8ed8cp-2}, + {-0x1.5b362cb974183p-57, -0x1.b5d1009e15ccp-2}, + {0x1.e97af1a63c807p-54, -0x1.cc66e9931c46p-2}, + {0x1.c3e4edc5872f8p-55, -0x1.e2b5d3806f63cp-2}, + {-0x1.fb44f80f92225p-54, -0x1.f8ba4dbf89ab8p-2}, + {-0x1.9697faf2e2fe5p-53, -0x1.073879922ffecp-1}, + {0x1.7bc8eda6af93cp-53, -0x1.11eb3541b4b24p-1}, + {-0x1.b25dd267f66p-55, -0x1.1c73b39ae68c8p-1}, + {0x1.5769d0fbcddc3p-53, -0x1.26d054cdd12ep-1}, + {-0x1.c20673b2116b2p-54, -0x1.30ff7fce17034p-1}, + {-0x1.3c7c4bc72a92cp-53, -0x1.3affa292050b8p-1}, + {0x1.e7f895d302395p-53, -0x1.44cf325091dd8p-1}, + {-0x1.13c293edceb32p-53, -0x1.4e6cabbe3e5e8p-1}, + {0x1.75720992bfbb2p-55, -0x1.57d69348cecap-1}, + {0x1.24a366a5fe547p-53, -0x1.610b7551d2cep-1}, + {-0x1.21165f626cdd5p-54, -0x1.6a09e667f3bccp-1}, + {0x1.bcac43c389ba9p-53, -0x1.72d0837efff98p-1}, + {0x1.21ea6f59be15bp-53, -0x1.7b5df226aafbp-1}, + {-0x1.d217be0e2b971p-53, -0x1.83b0e0bff976cp-1}, + {-0x1.69d0f6897664ap-54, -0x1.8bc806b15174p-1}, + {0x1.615f32b6f907ap-54, -0x1.93a22499263fcp-1}, + {-0x1.6788ebcc76dc6p-54, -0x1.9b3e047f3874p-1}, + {-0x1.ddae89fd441d1p-53, -0x1.a29a7a046278p-1}, + {0x1.f98273c5d2495p-54, -0x1.a9b66290ea1a4p-1}, + {0x1.926da300ffccep-55, -0x1.b090a581502p-1}, + {-0x1.90e58336c64a8p-53, -0x1.b728345196e3cp-1}, + {-0x1.9f6963354e3fep-53, -0x1.bd7c0ac6f9528p-1}, + {-0x1.a47d3a2a0dcbep-54, -0x1.c38b2f180bdbp-1}, + {-0x1.ed0489e16b9ap-54, -0x1.c954b213411f4p-1}, + {0x1.0f3db5dad5ac5p-53, -0x1.ced7af43cc774p-1}, + {-0x1.ac42b5a8b6943p-53, -0x1.d4134d14dc938p-1}, + {0x1.d75033dfb9ca8p-53, -0x1.d906bcf328d48p-1}, + {-0x1.83c37c6107db3p-55, -0x1.ddb13b6ccc23cp-1}, + {-0x1.7f59c49f6cd6dp-54, -0x1.e212104f686e4p-1}, + {-0x1.ee94a90d7b88bp-53, -0x1.e6288ec48e11p-1}, + {0x1.a27d3874701f9p-53, -0x1.e9f4156c62ddcp-1}, + {0x1.85f4e1b8298dp-54, -0x1.ed740e7684964p-1}, + {0x1.ab4e148e52d9ep-54, -0x1.f0a7efb9230d8p-1}, + {-0x1.8a11412b82346p-54, -0x1.f38f3ac64e588p-1}, + {-0x1.562172a361fd3p-56, -0x1.f6297cff75cbp-1}, + {-0x1.3564acef1ff97p-53, -0x1.f8764fa714ba8p-1}, + {0x1.5e82a3284d5c8p-53, -0x1.fa7557f08a518p-1}, + {0x1.709bccb89a989p-54, -0x1.fc26470e19fd4p-1}, + {-0x1.9e082721dfb8ep-53, -0x1.fd88da3d12524p-1}, + {0x1.eade132f3981dp-53, -0x1.fe9cdad01883cp-1}, + {-0x1.e3a843d1db55fp-53, -0x1.ff621e3796d7cp-1}, + {-0x1.765595d548d9ap-54, -0x1.ffd886084cd0cp-1}, + {0, -1}, + {-0x1.765595d548d9ap-54, -0x1.ffd886084cd0cp-1}, + {-0x1.e3a843d1db55fp-53, -0x1.ff621e3796d7cp-1}, + {0x1.eade132f3981dp-53, -0x1.fe9cdad01883cp-1}, + {-0x1.9e082721dfb8ep-53, -0x1.fd88da3d12524p-1}, + {0x1.709bccb89a989p-54, -0x1.fc26470e19fd4p-1}, + {0x1.5e82a3284d5c8p-53, -0x1.fa7557f08a518p-1}, + {-0x1.3564acef1ff97p-53, -0x1.f8764fa714ba8p-1}, + {-0x1.562172a361fd3p-56, -0x1.f6297cff75cbp-1}, + {-0x1.8a11412b82346p-54, -0x1.f38f3ac64e588p-1}, + {0x1.ab4e148e52d9ep-54, -0x1.f0a7efb9230d8p-1}, + {0x1.85f4e1b8298dp-54, -0x1.ed740e7684964p-1}, + {0x1.a27d3874701f9p-53, -0x1.e9f4156c62ddcp-1}, + {-0x1.ee94a90d7b88bp-53, -0x1.e6288ec48e11p-1}, + {-0x1.7f59c49f6cd6dp-54, -0x1.e212104f686e4p-1}, + {-0x1.83c37c6107db3p-55, -0x1.ddb13b6ccc23cp-1}, + {0x1.d75033dfb9ca8p-53, -0x1.d906bcf328d48p-1}, + {-0x1.ac42b5a8b6943p-53, -0x1.d4134d14dc938p-1}, + {0x1.0f3db5dad5ac5p-53, -0x1.ced7af43cc774p-1}, + {-0x1.ed0489e16b9ap-54, -0x1.c954b213411f4p-1}, + {-0x1.a47d3a2a0dcbep-54, -0x1.c38b2f180bdbp-1}, + {-0x1.9f6963354e3fep-53, -0x1.bd7c0ac6f9528p-1}, + {-0x1.90e58336c64a8p-53, -0x1.b728345196e3cp-1}, + {0x1.926da300ffccep-55, -0x1.b090a581502p-1}, + {0x1.f98273c5d2495p-54, -0x1.a9b66290ea1a4p-1}, + {-0x1.ddae89fd441d1p-53, -0x1.a29a7a046278p-1}, + {-0x1.6788ebcc76dc6p-54, -0x1.9b3e047f3874p-1}, + {0x1.615f32b6f907ap-54, -0x1.93a22499263fcp-1}, + {-0x1.69d0f6897664ap-54, -0x1.8bc806b15174p-1}, + {-0x1.d217be0e2b971p-53, -0x1.83b0e0bff976cp-1}, + {0x1.21ea6f59be15bp-53, -0x1.7b5df226aafbp-1}, + {0x1.bcac43c389ba9p-53, -0x1.72d0837efff98p-1}, + {-0x1.21165f626cdd5p-54, -0x1.6a09e667f3bccp-1}, + {0x1.24a366a5fe547p-53, -0x1.610b7551d2cep-1}, + {0x1.75720992bfbb2p-55, -0x1.57d69348cecap-1}, + {-0x1.13c293edceb32p-53, -0x1.4e6cabbe3e5e8p-1}, + {0x1.e7f895d302395p-53, -0x1.44cf325091dd8p-1}, + {-0x1.3c7c4bc72a92cp-53, -0x1.3affa292050b8p-1}, + {-0x1.c20673b2116b2p-54, -0x1.30ff7fce17034p-1}, + {0x1.5769d0fbcddc3p-53, -0x1.26d054cdd12ep-1}, + {-0x1.b25dd267f66p-55, -0x1.1c73b39ae68c8p-1}, + {0x1.7bc8eda6af93cp-53, -0x1.11eb3541b4b24p-1}, + {-0x1.9697faf2e2fe5p-53, -0x1.073879922ffecp-1}, + {-0x1.fb44f80f92225p-54, -0x1.f8ba4dbf89ab8p-2}, + {0x1.c3e4edc5872f8p-55, -0x1.e2b5d3806f63cp-2}, + {0x1.e97af1a63c807p-54, -0x1.cc66e9931c46p-2}, + {-0x1.5b362cb974183p-57, -0x1.b5d1009e15ccp-2}, + {0x1.d24afdade848bp-54, -0x1.9ef7943a8ed8cp-2}, + {0x1.2e59dba7ab4c2p-54, -0x1.87de2a6aea964p-2}, + {0x1.512c678219317p-54, -0x1.7088530fa45ap-2}, + {-0x1.fc2047e54e614p-55, -0x1.58f9a75ab1fdcp-2}, + {-0x1.4325f12be8946p-54, -0x1.4135c941766p-2}, + {-0x1.a8b5c974ee7b5p-54, -0x1.294062ed59f04p-2}, + {0x1.3ed9efaa42ab3p-55, -0x1.111d262b1f678p-2}, + {0x1.50b7bbc4768b1p-55, -0x1.f19f97b215f1cp-3}, + {0x1.035e2873ca432p-55, -0x1.c0b826a7e4f64p-3}, + {0x1.49b466e7fe36p-55, -0x1.8f8b83c69a60cp-3}, + {0x1.ab3802218894fp-55, -0x1.5e214448b3fc8p-3}, + {0x1.dd9ffeaecbdc4p-55, -0x1.2c8106e8e613cp-3}, + {0x1.cbb1f71aca352p-56, -0x1.f564e56a9731p-4}, + {0x1.e2718d26ed688p-60, -0x1.917a6bc29b42cp-4}, + {-0x1.ccbeeeae8129ap-56, -0x1.2d52092ce19f4p-4}, + {0x1.912bd0d569a9p-61, -0x1.91f65f10dd814p-5}, + {-0x1.f938a73db97fbp-58, -0x1.92155f7a3667cp-6}, +}; + +LIBC_INLINE unsigned range_reduction_small(double x, DoubleDouble &u) { + constexpr double ONE_TWENTY_EIGHT_OVER_PI = 0x1.45f306dc9c883p5; + + // Digits of -pi/128, generated by Sollya with: + // > a = round(-pi/128, 25, RN); + // > b = round(-pi/128 - a, 23, RN); + // > c = round(-pi/128 - a - b, 25, RN); + // > d = round(-pi/128 - a - b - c, D, RN); + // -pi/128 ~ a + b + c + d + // The precisions of the parts are chosen so that: + // 1) k * a, k * b, k * c are exact in double precision + // 2) k * b + (x - (k * a)) is exact in double precsion + constexpr double MPI_OVER_128[4] = {-0x1.921fb5p-6, -0x1.110b48p-32, + +0x1.ee59dap-56, -0x1.98a2e03707345p-83}; + + double prod_hi = x * ONE_TWENTY_EIGHT_OVER_PI; + double kd = fputil::nearest_integer(prod_hi); + + // With -pi/128 ~ a + b + c + d as in MPI_OVER_128 description: + // t = x + k * a + double t = fputil::multiply_add(kd, MPI_OVER_128[0], x); // Exact + // y_hi = t + k * b = (x + k * a) + k * b + double y_hi = fputil::multiply_add(kd, MPI_OVER_128[1], t); // Exact + // y_lo ~ k * c + k * d + double y_lo = fputil::multiply_add(kd, MPI_OVER_128[2], kd * MPI_OVER_128[3]); + // u.hi + u.lo ~ x + k * (a + b + c + d) + u = fputil::exact_add(y_hi, y_lo); + // Error bound: For |x| < 2^-23, + // |(x mod pi/128) - (u_hi + u_lo)| < ulp(y_lo) + // <= ulp(2 * x * c) + // <= ulp(2^24 * 2^-56) + // = 2^(24 - 56 - 52) + // = 2^-84 + return static_cast<unsigned>(static_cast<int>(kd)); +} + +} // namespace nofma + +} // namespace LIBC_NAMESPACE + +#endif // LLVM_LIBC_SRC_MATH_GENERIC_RANGE_REDUCTION_DOUBLE_NOFMA_H |