Index: head/lib/msun/ld80/k_expl.h =================================================================== --- head/lib/msun/ld80/k_expl.h (revision 308171) +++ head/lib/msun/ld80/k_expl.h (revision 308172) @@ -1,305 +1,305 @@ /* from: FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl */ /*- * Copyright (c) 2009-2013 Steven G. Kargl * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice unmodified, this list of conditions, and the following * disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. * * Optimized by Bruce D. Evans. */ #include __FBSDID("$FreeBSD$"); /* * See s_expl.c for more comments about __k_expl(). * * See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments * about the secondary kernels. */ #define INTERVALS 128 #define LOG2_INTERVALS 7 #define BIAS (LDBL_MAX_EXP - 1) static const double /* * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest * bits zero so that multiplication of it by n is exact. */ INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */ L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */ /* * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]: * |exp(x) - p(x)| < 2**-77.2 * (0.002708 is ln2/(2*INTERVALS) rounded up a little). */ A2 = 0.5, A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */ A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */ A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */ A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */ /* * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where * the first 53 bits of the significand are stored in hi and the next 53 * bits are in lo. Tang's paper states that the trailing 6 bits of hi must * be zero for his algorithm in both single and double precision, because * the table is re-used in the implementation of expm1() where a floating * point addition involving hi must be exact. Here hi is double, so * converting it to long double gives 11 trailing zero bits. */ static const struct { double hi; double lo; } tbl[INTERVALS] = { - 0x1p+0, 0x0p+0, + { 0x1p+0, 0x0p+0 }, /* * XXX hi is rounded down, and the formatting is not quite normal. * But I rather like both. The 0x1.*p format is good for 4N+1 * mantissa bits. Rounding down makes the lo terms positive, * so that the columnar formatting can be simpler. */ - 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54, - 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53, - 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53, - 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55, - 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53, - 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57, - 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54, - 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54, - 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54, - 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59, - 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53, - 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53, - 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53, - 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53, - 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55, - 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53, - 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53, - 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55, - 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53, - 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54, - 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53, - 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55, - 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55, - 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54, - 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55, - 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55, - 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53, - 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55, - 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53, - 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54, - 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56, - 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55, - 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55, - 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54, - 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53, - 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53, - 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53, - 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53, - 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53, - 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55, - 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53, - 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53, - 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53, - 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59, - 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54, - 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56, - 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54, - 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56, - 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54, - 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53, - 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53, - 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53, - 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53, - 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54, - 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55, - 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54, - 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60, - 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54, - 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53, - 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53, - 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53, - 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53, - 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57, - 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53, - 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53, - 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53, - 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53, - 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53, - 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53, - 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53, - 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54, - 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53, - 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54, - 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56, - 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53, - 0x1.82589994cce12p+0, 0x1.159f115f56694p-53, - 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53, - 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53, - 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54, - 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54, - 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53, - 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55, - 0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53, - 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53, - 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53, - 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53, - 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53, - 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56, - 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56, - 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53, - 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54, - 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53, - 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54, - 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54, - 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53, - 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54, - 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53, - 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53, - 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53, - 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53, - 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53, - 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55, - 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53, - 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55, - 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54, - 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54, - 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56, - 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56, - 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53, - 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53, - 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53, - 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55, - 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53, - 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54, - 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54, - 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53, - 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53, - 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55, - 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54, - 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53, - 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53, - 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54, - 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54, - 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54, - 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53, - 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55, - 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 + { 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54 }, + { 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53 }, + { 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53 }, + { 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55 }, + { 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53 }, + { 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57 }, + { 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54 }, + { 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54 }, + { 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54 }, + { 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59 }, + { 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53 }, + { 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53 }, + { 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53 }, + { 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53 }, + { 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55 }, + { 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53 }, + { 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53 }, + { 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55 }, + { 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53 }, + { 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54 }, + { 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53 }, + { 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55 }, + { 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55 }, + { 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54 }, + { 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55 }, + { 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55 }, + { 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53 }, + { 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55 }, + { 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53 }, + { 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54 }, + { 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56 }, + { 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55 }, + { 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55 }, + { 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54 }, + { 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53 }, + { 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53 }, + { 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53 }, + { 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53 }, + { 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53 }, + { 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55 }, + { 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53 }, + { 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53 }, + { 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53 }, + { 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59 }, + { 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54 }, + { 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56 }, + { 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54 }, + { 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56 }, + { 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54 }, + { 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53 }, + { 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53 }, + { 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53 }, + { 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53 }, + { 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54 }, + { 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55 }, + { 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54 }, + { 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60 }, + { 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54 }, + { 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53 }, + { 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53 }, + { 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53 }, + { 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53 }, + { 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57 }, + { 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53 }, + { 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53 }, + { 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53 }, + { 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53 }, + { 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53 }, + { 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53 }, + { 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53 }, + { 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54 }, + { 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53 }, + { 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54 }, + { 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56 }, + { 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53 }, + { 0x1.82589994cce12p+0, 0x1.159f115f56694p-53 }, + { 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53 }, + { 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53 }, + { 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54 }, + { 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54 }, + { 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53 }, + { 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55 }, + { 0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53 }, + { 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53 }, + { 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53 }, + { 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53 }, + { 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53 }, + { 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56 }, + { 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56 }, + { 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53 }, + { 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54 }, + { 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53 }, + { 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54 }, + { 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54 }, + { 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53 }, + { 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54 }, + { 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53 }, + { 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53 }, + { 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53 }, + { 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53 }, + { 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53 }, + { 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55 }, + { 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53 }, + { 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55 }, + { 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54 }, + { 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54 }, + { 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56 }, + { 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56 }, + { 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53 }, + { 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53 }, + { 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53 }, + { 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55 }, + { 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53 }, + { 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54 }, + { 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54 }, + { 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53 }, + { 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53 }, + { 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55 }, + { 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54 }, + { 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53 }, + { 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53 }, + { 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54 }, + { 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54 }, + { 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54 }, + { 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53 }, + { 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55 }, + { 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 } }; /* * Kernel for expl(x). x must be finite and not tiny or huge. * "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN). * "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2). */ static inline void __k_expl(long double x, long double *hip, long double *lop, int *kp) { long double fn, q, r, r1, r2, t, z; int n, n2; /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ /* Use a specialized rint() to get fn. Assume round-to-nearest. */ fn = x * INV_L + 0x1.8p63 - 0x1.8p63; r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */ #if defined(HAVE_EFFICIENT_IRINTL) n = irintl(fn); #elif defined(HAVE_EFFICIENT_IRINT) n = irint(fn); #else n = (int)fn; #endif n2 = (unsigned)n % INTERVALS; /* Depend on the sign bit being propagated: */ *kp = n >> LOG2_INTERVALS; r1 = x - fn * L1; r2 = fn * -L2; /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ z = r * r; #if 0 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; #else q = r2 + z * A2 + z * r * (A3 + r * A4 + z * (A5 + r * A6)); #endif t = (long double)tbl[n2].lo + tbl[n2].hi; *hip = tbl[n2].hi; *lop = tbl[n2].lo + t * (q + r1); } static inline void k_hexpl(long double x, long double *hip, long double *lop) { float twopkm1; int k; __k_expl(x, hip, lop, &k); SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23)); *hip *= twopkm1; *lop *= twopkm1; } static inline long double hexpl(long double x) { long double hi, lo, twopkm2; int k; twopkm2 = 1; __k_expl(x, &hi, &lo, &k); SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2); return (lo + hi) * 2 * twopkm2; } #ifdef _COMPLEX_H /* * See ../src/k_exp.c for details. */ static inline long double complex __ldexp_cexpl(long double complex z, int expt) { long double exp_x, hi, lo; long double x, y, scale1, scale2; int half_expt, k; x = creall(z); y = cimagl(z); __k_expl(x, &hi, &lo, &k); exp_x = (lo + hi) * 0x1p16382; expt += k - 16382; scale1 = 1; half_expt = expt / 2; SET_LDBL_EXPSIGN(scale1, BIAS + half_expt); scale2 = 1; SET_LDBL_EXPSIGN(scale1, BIAS + expt - half_expt); return (CMPLXL(cos(y) * exp_x * scale1 * scale2, sinl(y) * exp_x * scale1 * scale2)); } #endif /* _COMPLEX_H */ Index: head/lib/msun/ld80/s_logl.c =================================================================== --- head/lib/msun/ld80/s_logl.c (revision 308171) +++ head/lib/msun/ld80/s_logl.c (revision 308172) @@ -1,717 +1,717 @@ /*- * Copyright (c) 2007-2013 Bruce D. Evans * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice unmodified, this list of conditions, and the following * disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ #include __FBSDID("$FreeBSD$"); /** * Implementation of the natural logarithm of x for Intel 80-bit format. * * First decompose x into its base 2 representation: * * log(x) = log(X * 2**k), where X is in [1, 2) * = log(X) + k * log(2). * * Let X = X_i + e, where X_i is the center of one of the intervals * [-1.0/256, 1.0/256), [1.0/256, 3.0/256), .... [2.0-1.0/256, 2.0+1.0/256) * and X is in this interval. Then * * log(X) = log(X_i + e) * = log(X_i * (1 + e / X_i)) * = log(X_i) + log(1 + e / X_i). * * The values log(X_i) are tabulated below. Let d = e / X_i and use * * log(1 + d) = p(d) * * where p(d) = d - 0.5*d*d + ... is a special minimax polynomial of * suitably high degree. * * To get sufficiently small roundoff errors, k * log(2), log(X_i), and * sometimes (if |k| is not large) the first term in p(d) must be evaluated * and added up in extra precision. Extra precision is not needed for the * rest of p(d). In the worst case when k = 0 and log(X_i) is 0, the final * error is controlled mainly by the error in the second term in p(d). The * error in this term itself is at most 0.5 ulps from the d*d operation in * it. The error in this term relative to the first term is thus at most * 0.5 * |-0.5| * |d| < 1.0/1024 ulps. We aim for an accumulated error of * at most twice this at the point of the final rounding step. Thus the * final error should be at most 0.5 + 1.0/512 = 0.5020 ulps. Exhaustive * testing of a float variant of this function showed a maximum final error * of 0.5008 ulps. Non-exhaustive testing of a double variant of this * function showed a maximum final error of 0.5078 ulps (near 1+1.0/256). * * We made the maximum of |d| (and thus the total relative error and the * degree of p(d)) small by using a large number of intervals. Using * centers of intervals instead of endpoints reduces this maximum by a * factor of 2 for a given number of intervals. p(d) is special only * in beginning with the Taylor coefficients 0 + 1*d, which tends to happen * naturally. The most accurate minimax polynomial of a given degree might * be different, but then we wouldn't want it since we would have to do * extra work to avoid roundoff error (especially for P0*d instead of d). */ #ifdef DEBUG #include #include #endif #ifdef __i386__ #include #endif #include "fpmath.h" #include "math.h" #define i386_SSE_GOOD #ifndef NO_STRUCT_RETURN #define STRUCT_RETURN #endif #include "math_private.h" #if !defined(NO_UTAB) && !defined(NO_UTABL) #define USE_UTAB #endif /* * Domain [-0.005280, 0.004838], range ~[-5.1736e-22, 5.1738e-22]: * |log(1 + d)/d - p(d)| < 2**-70.7 */ static const double P2 = -0.5, P3 = 3.3333333333333359e-1, /* 0x1555555555555a.0p-54 */ P4 = -2.5000000000004424e-1, /* -0x1000000000031d.0p-54 */ P5 = 1.9999999992970016e-1, /* 0x1999999972f3c7.0p-55 */ P6 = -1.6666666072191585e-1, /* -0x15555548912c09.0p-55 */ P7 = 1.4286227413310518e-1, /* 0x12494f9d9def91.0p-55 */ P8 = -1.2518388626763144e-1; /* -0x1006068cc0b97c.0p-55 */ static volatile const double zero = 0; #define INTERVALS 128 #define LOG2_INTERVALS 7 #define TSIZE (INTERVALS + 1) #define G(i) (T[(i)].G) #define F_hi(i) (T[(i)].F_hi) #define F_lo(i) (T[(i)].F_lo) #define ln2_hi F_hi(TSIZE - 1) #define ln2_lo F_lo(TSIZE - 1) #define E(i) (U[(i)].E) #define H(i) (U[(i)].H) static const struct { float G; /* 1/(1 + i/128) rounded to 8/9 bits */ float F_hi; /* log(1 / G_i) rounded (see below) */ double F_lo; /* next 53 bits for log(1 / G_i) */ } T[TSIZE] = { /* * ln2_hi and each F_hi(i) are rounded to a number of bits that * makes F_hi(i) + dk*ln2_hi exact for all i and all dk. * * The last entry (for X just below 2) is used to define ln2_hi * and ln2_lo, to ensure that F_hi(i) and F_lo(i) cancel exactly * with dk*ln2_hi and dk*ln2_lo, respectively, when dk = -1. * This is needed for accuracy when x is just below 1. (To avoid * special cases, such x are "reduced" strangely to X just below * 2 and dk = -1, and then the exact cancellation is needed * because any the error from any non-exactness would be too * large). * * We want to share this table between double precision and ld80, * so the relevant range of dk is the larger one of ld80 * ([-16445, 16383]) and the relevant exactness requirement is * the stricter one of double precision. The maximum number of * bits in F_hi(i) that works is very dependent on i but has * a minimum of 33. We only need about 12 bits in F_hi(i) for * it to provide enough extra precision in double precision (11 * more than that are required for ld80). * * We round F_hi(i) to 24 bits so that it can have type float, * mainly to minimize the size of the table. Using all 24 bits * in a float for it automatically satisfies the above constraints. */ - 0x800000.0p-23, 0, 0, - 0xfe0000.0p-24, 0x8080ac.0p-30, -0x14ee431dae6675.0p-84, - 0xfc0000.0p-24, 0x8102b3.0p-29, -0x1db29ee2d83718.0p-84, - 0xfa0000.0p-24, 0xc24929.0p-29, 0x1191957d173698.0p-83, - 0xf80000.0p-24, 0x820aec.0p-28, 0x13ce8888e02e79.0p-82, - 0xf60000.0p-24, 0xa33577.0p-28, -0x17a4382ce6eb7c.0p-82, - 0xf48000.0p-24, 0xbc42cb.0p-28, -0x172a21161a1076.0p-83, - 0xf30000.0p-24, 0xd57797.0p-28, -0x1e09de07cb9589.0p-82, - 0xf10000.0p-24, 0xf7518e.0p-28, 0x1ae1eec1b036c5.0p-91, - 0xef0000.0p-24, 0x8cb9df.0p-27, -0x1d7355325d560e.0p-81, - 0xed8000.0p-24, 0x999ec0.0p-27, -0x1f9f02d256d503.0p-82, - 0xec0000.0p-24, 0xa6988b.0p-27, -0x16fc0a9d12c17a.0p-83, - 0xea0000.0p-24, 0xb80698.0p-27, 0x15d581c1e8da9a.0p-81, - 0xe80000.0p-24, 0xc99af3.0p-27, -0x1535b3ba8f150b.0p-83, - 0xe70000.0p-24, 0xd273b2.0p-27, 0x163786f5251af0.0p-85, - 0xe50000.0p-24, 0xe442c0.0p-27, 0x1bc4b2368e32d5.0p-84, - 0xe38000.0p-24, 0xf1b83f.0p-27, 0x1c6090f684e676.0p-81, - 0xe20000.0p-24, 0xff448a.0p-27, -0x1890aa69ac9f42.0p-82, - 0xe08000.0p-24, 0x8673f6.0p-26, 0x1b9985194b6b00.0p-80, - 0xdf0000.0p-24, 0x8d515c.0p-26, -0x1dc08d61c6ef1e.0p-83, - 0xdd8000.0p-24, 0x943a9e.0p-26, -0x1f72a2dac729b4.0p-82, - 0xdc0000.0p-24, 0x9b2fe6.0p-26, -0x1fd4dfd3a0afb9.0p-80, - 0xda8000.0p-24, 0xa2315d.0p-26, -0x11b26121629c47.0p-82, - 0xd90000.0p-24, 0xa93f2f.0p-26, 0x1286d633e8e569.0p-81, - 0xd78000.0p-24, 0xb05988.0p-26, 0x16128eba936770.0p-84, - 0xd60000.0p-24, 0xb78094.0p-26, 0x16ead577390d32.0p-80, - 0xd50000.0p-24, 0xbc4c6c.0p-26, 0x151131ccf7c7b7.0p-81, - 0xd38000.0p-24, 0xc3890a.0p-26, -0x115e2cd714bd06.0p-80, - 0xd20000.0p-24, 0xcad2d7.0p-26, -0x1847f406ebd3b0.0p-82, - 0xd10000.0p-24, 0xcfb620.0p-26, 0x1c2259904d6866.0p-81, - 0xcf8000.0p-24, 0xd71653.0p-26, 0x1ece57a8d5ae55.0p-80, - 0xce0000.0p-24, 0xde843a.0p-26, -0x1f109d4bc45954.0p-81, - 0xcd0000.0p-24, 0xe37fde.0p-26, 0x1bc03dc271a74d.0p-81, - 0xcb8000.0p-24, 0xeb050c.0p-26, -0x1bf2badc0df842.0p-85, - 0xca0000.0p-24, 0xf29878.0p-26, -0x18efededd89fbe.0p-87, - 0xc90000.0p-24, 0xf7ad6f.0p-26, 0x1373ff977baa69.0p-81, - 0xc80000.0p-24, 0xfcc8e3.0p-26, 0x196766f2fb3283.0p-80, - 0xc68000.0p-24, 0x823f30.0p-25, 0x19bd076f7c434e.0p-79, - 0xc58000.0p-24, 0x84d52c.0p-25, -0x1a327257af0f46.0p-79, - 0xc40000.0p-24, 0x88bc74.0p-25, 0x113f23def19c5a.0p-81, - 0xc30000.0p-24, 0x8b5ae6.0p-25, 0x1759f6e6b37de9.0p-79, - 0xc20000.0p-24, 0x8dfccb.0p-25, 0x1ad35ca6ed5148.0p-81, - 0xc10000.0p-24, 0x90a22b.0p-25, 0x1a1d71a87deba4.0p-79, - 0xbf8000.0p-24, 0x94a0d8.0p-25, -0x139e5210c2b731.0p-80, - 0xbe8000.0p-24, 0x974f16.0p-25, -0x18f6ebcff3ed73.0p-81, - 0xbd8000.0p-24, 0x9a00f1.0p-25, -0x1aa268be39aab7.0p-79, - 0xbc8000.0p-24, 0x9cb672.0p-25, -0x14c8815839c566.0p-79, - 0xbb0000.0p-24, 0xa0cda1.0p-25, 0x1eaf46390dbb24.0p-81, - 0xba0000.0p-24, 0xa38c6e.0p-25, 0x138e20d831f698.0p-81, - 0xb90000.0p-24, 0xa64f05.0p-25, -0x1e8d3c41123616.0p-82, - 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0x990000.0p-24, 0x83c5f8.0p-24, 0x14cf15a048907b.0p-79 }, + { 0x988000.0p-24, 0x849c7d.0p-24, 0x1cbb1d35fb8287.0p-78 }, + { 0x978000.0p-24, 0x864ba6.0p-24, 0x1128639b814f9c.0p-78 }, + { 0x970000.0p-24, 0x87244c.0p-24, 0x184733853300f0.0p-79 }, + { 0x968000.0p-24, 0x87fdaa.0p-24, 0x109d23aef77dd6.0p-80 }, + { 0x958000.0p-24, 0x89b293.0p-24, -0x1a81ef367a59de.0p-78 }, + { 0x950000.0p-24, 0x8a8e20.0p-24, -0x121ad3dbb2f452.0p-78 }, + { 0x948000.0p-24, 0x8b6a6a.0p-24, -0x1cfb981628af72.0p-79 }, + { 0x938000.0p-24, 0x8d253a.0p-24, -0x1d21730ea76cfe.0p-79 }, + { 0x930000.0p-24, 0x8e03c2.0p-24, 0x135cc00e566f77.0p-78 }, + { 0x928000.0p-24, 0x8ee30d.0p-24, -0x10fcb5df257a26.0p-80 }, + { 0x918000.0p-24, 0x90a3ee.0p-24, -0x16e171b15433d7.0p-79 }, + { 0x910000.0p-24, 0x918587.0p-24, -0x1d050da07f3237.0p-79 }, + { 0x908000.0p-24, 0x9267e7.0p-24, 0x1be03669a5268d.0p-79 }, + { 0x8f8000.0p-24, 0x942f04.0p-24, 0x10b28e0e26c337.0p-79 }, + { 0x8f0000.0p-24, 0x9513c3.0p-24, 0x1a1d820da57cf3.0p-78 }, + { 0x8e8000.0p-24, 0x95f950.0p-24, -0x19ef8f13ae3cf1.0p-79 }, + { 0x8e0000.0p-24, 0x96dfab.0p-24, -0x109e417a6e507c.0p-78 }, + { 0x8d0000.0p-24, 0x98aed2.0p-24, 0x10d01a2c5b0e98.0p-79 }, + { 0x8c8000.0p-24, 0x9997a2.0p-24, -0x1d6a50d4b61ea7.0p-78 }, + { 0x8c0000.0p-24, 0x9a8145.0p-24, 0x1b3b190b83f952.0p-78 }, + { 0x8b8000.0p-24, 0x9b6bbf.0p-24, 0x13a69fad7e7abe.0p-78 }, + { 0x8b0000.0p-24, 0x9c5711.0p-24, -0x11cd12316f576b.0p-78 }, + { 0x8a8000.0p-24, 0x9d433b.0p-24, 0x1c95c444b807a2.0p-79 }, + { 0x898000.0p-24, 0x9f1e22.0p-24, -0x1b9c224ea698c3.0p-79 }, + { 0x890000.0p-24, 0xa00ce1.0p-24, 0x125ca93186cf0f.0p-81 }, + { 0x888000.0p-24, 0xa0fc80.0p-24, -0x1ee38a7bc228b3.0p-79 }, + { 0x880000.0p-24, 0xa1ed00.0p-24, -0x1a0db876613d20.0p-78 }, + { 0x878000.0p-24, 0xa2de62.0p-24, 0x193224e8516c01.0p-79 }, + { 0x870000.0p-24, 0xa3d0a9.0p-24, 0x1fa28b4d2541ad.0p-79 }, + { 0x868000.0p-24, 0xa4c3d6.0p-24, 0x1c1b5760fb4572.0p-78 }, + { 0x858000.0p-24, 0xa6acea.0p-24, 0x1fed5d0f65949c.0p-80 }, + { 0x850000.0p-24, 0xa7a2d4.0p-24, 0x1ad270c9d74936.0p-80 }, + { 0x848000.0p-24, 0xa899ab.0p-24, 0x199ff15ce53266.0p-79 }, + { 0x840000.0p-24, 0xa99171.0p-24, 0x1a19e15ccc45d2.0p-79 }, + { 0x838000.0p-24, 0xaa8a28.0p-24, -0x121a14ec532b36.0p-80 }, + { 0x830000.0p-24, 0xab83d1.0p-24, 0x1aee319980bff3.0p-79 }, + { 0x828000.0p-24, 0xac7e6f.0p-24, -0x18ffd9e3900346.0p-80 }, + { 0x820000.0p-24, 0xad7a03.0p-24, -0x1e4db102ce29f8.0p-80 }, + { 0x818000.0p-24, 0xae768f.0p-24, 0x17c35c55a04a83.0p-81 }, + { 0x810000.0p-24, 0xaf7415.0p-24, 0x1448324047019b.0p-78 }, + { 0x808000.0p-24, 0xb07298.0p-24, -0x1750ee3915a198.0p-78 }, + { 0x800000.0p-24, 0xb17218.0p-24, -0x105c610ca86c39.0p-81 }, }; #ifdef USE_UTAB static const struct { float H; /* 1 + i/INTERVALS (exact) */ float E; /* H(i) * G(i) - 1 (exact) */ } U[TSIZE] = { - 0x800000.0p-23, 0, - 0x810000.0p-23, -0x800000.0p-37, - 0x820000.0p-23, -0x800000.0p-35, - 0x830000.0p-23, -0x900000.0p-34, - 0x840000.0p-23, -0x800000.0p-33, - 0x850000.0p-23, -0xc80000.0p-33, - 0x860000.0p-23, -0xa00000.0p-36, - 0x870000.0p-23, 0x940000.0p-33, - 0x880000.0p-23, 0x800000.0p-35, - 0x890000.0p-23, -0xc80000.0p-34, - 0x8a0000.0p-23, 0xe00000.0p-36, - 0x8b0000.0p-23, 0x900000.0p-33, - 0x8c0000.0p-23, -0x800000.0p-35, - 0x8d0000.0p-23, -0xe00000.0p-33, - 0x8e0000.0p-23, 0x880000.0p-33, - 0x8f0000.0p-23, -0xa80000.0p-34, - 0x900000.0p-23, -0x800000.0p-35, - 0x910000.0p-23, 0x800000.0p-37, - 0x920000.0p-23, 0x900000.0p-35, - 0x930000.0p-23, 0xd00000.0p-35, - 0x940000.0p-23, 0xe00000.0p-35, - 0x950000.0p-23, 0xc00000.0p-35, - 0x960000.0p-23, 0xe00000.0p-36, - 0x970000.0p-23, -0x800000.0p-38, - 0x980000.0p-23, -0xc00000.0p-35, - 0x990000.0p-23, -0xd00000.0p-34, - 0x9a0000.0p-23, 0x880000.0p-33, - 0x9b0000.0p-23, 0xe80000.0p-35, - 0x9c0000.0p-23, -0x800000.0p-35, - 0x9d0000.0p-23, 0xb40000.0p-33, - 0x9e0000.0p-23, 0x880000.0p-34, - 0x9f0000.0p-23, -0xe00000.0p-35, - 0xa00000.0p-23, 0x800000.0p-33, - 0xa10000.0p-23, -0x900000.0p-36, - 0xa20000.0p-23, -0xb00000.0p-33, - 0xa30000.0p-23, -0xa00000.0p-36, - 0xa40000.0p-23, 0x800000.0p-33, - 0xa50000.0p-23, -0xf80000.0p-35, - 0xa60000.0p-23, 0x880000.0p-34, - 0xa70000.0p-23, -0x900000.0p-33, - 0xa80000.0p-23, -0x800000.0p-35, - 0xa90000.0p-23, 0x900000.0p-34, - 0xaa0000.0p-23, 0xa80000.0p-33, - 0xab0000.0p-23, -0xac0000.0p-34, - 0xac0000.0p-23, -0x800000.0p-37, - 0xad0000.0p-23, 0xf80000.0p-35, - 0xae0000.0p-23, 0xf80000.0p-34, - 0xaf0000.0p-23, -0xac0000.0p-33, - 0xb00000.0p-23, -0x800000.0p-33, - 0xb10000.0p-23, -0xb80000.0p-34, - 0xb20000.0p-23, -0x800000.0p-34, - 0xb30000.0p-23, -0xb00000.0p-35, - 0xb40000.0p-23, -0x800000.0p-35, - 0xb50000.0p-23, -0xe00000.0p-36, - 0xb60000.0p-23, -0x800000.0p-35, - 0xb70000.0p-23, -0xb00000.0p-35, - 0xb80000.0p-23, -0x800000.0p-34, - 0xb90000.0p-23, -0xb80000.0p-34, - 0xba0000.0p-23, -0x800000.0p-33, - 0xbb0000.0p-23, -0xac0000.0p-33, - 0xbc0000.0p-23, 0x980000.0p-33, - 0xbd0000.0p-23, 0xbc0000.0p-34, - 0xbe0000.0p-23, 0xe00000.0p-36, - 0xbf0000.0p-23, -0xb80000.0p-35, - 0xc00000.0p-23, -0x800000.0p-33, - 0xc10000.0p-23, 0xa80000.0p-33, - 0xc20000.0p-23, 0x900000.0p-34, - 0xc30000.0p-23, -0x800000.0p-35, - 0xc40000.0p-23, -0x900000.0p-33, - 0xc50000.0p-23, 0x820000.0p-33, - 0xc60000.0p-23, 0x800000.0p-38, - 0xc70000.0p-23, -0x820000.0p-33, - 0xc80000.0p-23, 0x800000.0p-33, - 0xc90000.0p-23, -0xa00000.0p-36, - 0xca0000.0p-23, -0xb00000.0p-33, - 0xcb0000.0p-23, 0x840000.0p-34, - 0xcc0000.0p-23, -0xd00000.0p-34, - 0xcd0000.0p-23, 0x800000.0p-33, - 0xce0000.0p-23, -0xe00000.0p-35, - 0xcf0000.0p-23, 0xa60000.0p-33, - 0xd00000.0p-23, -0x800000.0p-35, - 0xd10000.0p-23, 0xb40000.0p-33, - 0xd20000.0p-23, -0x800000.0p-35, - 0xd30000.0p-23, 0xaa0000.0p-33, - 0xd40000.0p-23, -0xe00000.0p-35, - 0xd50000.0p-23, 0x880000.0p-33, - 0xd60000.0p-23, -0xd00000.0p-34, - 0xd70000.0p-23, 0x9c0000.0p-34, - 0xd80000.0p-23, -0xb00000.0p-33, - 0xd90000.0p-23, -0x800000.0p-38, - 0xda0000.0p-23, 0xa40000.0p-33, - 0xdb0000.0p-23, -0xdc0000.0p-34, - 0xdc0000.0p-23, 0xc00000.0p-35, - 0xdd0000.0p-23, 0xca0000.0p-33, - 0xde0000.0p-23, -0xb80000.0p-34, - 0xdf0000.0p-23, 0xd00000.0p-35, - 0xe00000.0p-23, 0xc00000.0p-33, - 0xe10000.0p-23, -0xf40000.0p-34, - 0xe20000.0p-23, 0x800000.0p-37, - 0xe30000.0p-23, 0x860000.0p-33, - 0xe40000.0p-23, -0xc80000.0p-33, - 0xe50000.0p-23, -0xa80000.0p-34, - 0xe60000.0p-23, 0xe00000.0p-36, - 0xe70000.0p-23, 0x880000.0p-33, - 0xe80000.0p-23, -0xe00000.0p-33, - 0xe90000.0p-23, -0xfc0000.0p-34, - 0xea0000.0p-23, -0x800000.0p-35, - 0xeb0000.0p-23, 0xe80000.0p-35, - 0xec0000.0p-23, 0x900000.0p-33, - 0xed0000.0p-23, 0xe20000.0p-33, - 0xee0000.0p-23, -0xac0000.0p-33, - 0xef0000.0p-23, -0xc80000.0p-34, - 0xf00000.0p-23, -0x800000.0p-35, - 0xf10000.0p-23, 0x800000.0p-35, - 0xf20000.0p-23, 0xb80000.0p-34, - 0xf30000.0p-23, 0x940000.0p-33, - 0xf40000.0p-23, 0xc80000.0p-33, - 0xf50000.0p-23, -0xf20000.0p-33, - 0xf60000.0p-23, -0xc80000.0p-33, - 0xf70000.0p-23, -0xa20000.0p-33, - 0xf80000.0p-23, -0x800000.0p-33, - 0xf90000.0p-23, -0xc40000.0p-34, - 0xfa0000.0p-23, -0x900000.0p-34, - 0xfb0000.0p-23, -0xc80000.0p-35, - 0xfc0000.0p-23, -0x800000.0p-35, - 0xfd0000.0p-23, -0x900000.0p-36, - 0xfe0000.0p-23, -0x800000.0p-37, - 0xff0000.0p-23, -0x800000.0p-39, - 0x800000.0p-22, 0, + { 0x800000.0p-23, 0 }, + { 0x810000.0p-23, -0x800000.0p-37 }, + { 0x820000.0p-23, -0x800000.0p-35 }, + { 0x830000.0p-23, -0x900000.0p-34 }, + { 0x840000.0p-23, -0x800000.0p-33 }, + { 0x850000.0p-23, -0xc80000.0p-33 }, + { 0x860000.0p-23, -0xa00000.0p-36 }, + { 0x870000.0p-23, 0x940000.0p-33 }, + { 0x880000.0p-23, 0x800000.0p-35 }, + { 0x890000.0p-23, -0xc80000.0p-34 }, + { 0x8a0000.0p-23, 0xe00000.0p-36 }, + { 0x8b0000.0p-23, 0x900000.0p-33 }, + { 0x8c0000.0p-23, -0x800000.0p-35 }, + { 0x8d0000.0p-23, -0xe00000.0p-33 }, + { 0x8e0000.0p-23, 0x880000.0p-33 }, + { 0x8f0000.0p-23, -0xa80000.0p-34 }, + { 0x900000.0p-23, -0x800000.0p-35 }, + { 0x910000.0p-23, 0x800000.0p-37 }, + { 0x920000.0p-23, 0x900000.0p-35 }, + { 0x930000.0p-23, 0xd00000.0p-35 }, + { 0x940000.0p-23, 0xe00000.0p-35 }, + { 0x950000.0p-23, 0xc00000.0p-35 }, + { 0x960000.0p-23, 0xe00000.0p-36 }, + { 0x970000.0p-23, -0x800000.0p-38 }, + { 0x980000.0p-23, -0xc00000.0p-35 }, + { 0x990000.0p-23, -0xd00000.0p-34 }, + { 0x9a0000.0p-23, 0x880000.0p-33 }, + { 0x9b0000.0p-23, 0xe80000.0p-35 }, + { 0x9c0000.0p-23, -0x800000.0p-35 }, + { 0x9d0000.0p-23, 0xb40000.0p-33 }, + { 0x9e0000.0p-23, 0x880000.0p-34 }, + { 0x9f0000.0p-23, -0xe00000.0p-35 }, + { 0xa00000.0p-23, 0x800000.0p-33 }, + { 0xa10000.0p-23, -0x900000.0p-36 }, + { 0xa20000.0p-23, -0xb00000.0p-33 }, + { 0xa30000.0p-23, -0xa00000.0p-36 }, + { 0xa40000.0p-23, 0x800000.0p-33 }, + { 0xa50000.0p-23, -0xf80000.0p-35 }, + { 0xa60000.0p-23, 0x880000.0p-34 }, + { 0xa70000.0p-23, -0x900000.0p-33 }, + { 0xa80000.0p-23, -0x800000.0p-35 }, + { 0xa90000.0p-23, 0x900000.0p-34 }, + { 0xaa0000.0p-23, 0xa80000.0p-33 }, + { 0xab0000.0p-23, -0xac0000.0p-34 }, + { 0xac0000.0p-23, -0x800000.0p-37 }, + { 0xad0000.0p-23, 0xf80000.0p-35 }, + { 0xae0000.0p-23, 0xf80000.0p-34 }, + { 0xaf0000.0p-23, -0xac0000.0p-33 }, + { 0xb00000.0p-23, -0x800000.0p-33 }, + { 0xb10000.0p-23, -0xb80000.0p-34 }, + { 0xb20000.0p-23, -0x800000.0p-34 }, + { 0xb30000.0p-23, -0xb00000.0p-35 }, + { 0xb40000.0p-23, -0x800000.0p-35 }, + { 0xb50000.0p-23, -0xe00000.0p-36 }, + { 0xb60000.0p-23, -0x800000.0p-35 }, + { 0xb70000.0p-23, -0xb00000.0p-35 }, + { 0xb80000.0p-23, -0x800000.0p-34 }, + { 0xb90000.0p-23, -0xb80000.0p-34 }, + { 0xba0000.0p-23, -0x800000.0p-33 }, + { 0xbb0000.0p-23, -0xac0000.0p-33 }, + { 0xbc0000.0p-23, 0x980000.0p-33 }, + { 0xbd0000.0p-23, 0xbc0000.0p-34 }, + { 0xbe0000.0p-23, 0xe00000.0p-36 }, + { 0xbf0000.0p-23, -0xb80000.0p-35 }, + { 0xc00000.0p-23, -0x800000.0p-33 }, + { 0xc10000.0p-23, 0xa80000.0p-33 }, + { 0xc20000.0p-23, 0x900000.0p-34 }, + { 0xc30000.0p-23, -0x800000.0p-35 }, + { 0xc40000.0p-23, -0x900000.0p-33 }, + { 0xc50000.0p-23, 0x820000.0p-33 }, + { 0xc60000.0p-23, 0x800000.0p-38 }, + { 0xc70000.0p-23, -0x820000.0p-33 }, + { 0xc80000.0p-23, 0x800000.0p-33 }, + { 0xc90000.0p-23, -0xa00000.0p-36 }, + { 0xca0000.0p-23, -0xb00000.0p-33 }, + { 0xcb0000.0p-23, 0x840000.0p-34 }, + { 0xcc0000.0p-23, -0xd00000.0p-34 }, + { 0xcd0000.0p-23, 0x800000.0p-33 }, + { 0xce0000.0p-23, -0xe00000.0p-35 }, + { 0xcf0000.0p-23, 0xa60000.0p-33 }, + { 0xd00000.0p-23, -0x800000.0p-35 }, + { 0xd10000.0p-23, 0xb40000.0p-33 }, + { 0xd20000.0p-23, -0x800000.0p-35 }, + { 0xd30000.0p-23, 0xaa0000.0p-33 }, + { 0xd40000.0p-23, -0xe00000.0p-35 }, + { 0xd50000.0p-23, 0x880000.0p-33 }, + { 0xd60000.0p-23, -0xd00000.0p-34 }, + { 0xd70000.0p-23, 0x9c0000.0p-34 }, + { 0xd80000.0p-23, -0xb00000.0p-33 }, + { 0xd90000.0p-23, -0x800000.0p-38 }, + { 0xda0000.0p-23, 0xa40000.0p-33 }, + { 0xdb0000.0p-23, -0xdc0000.0p-34 }, + { 0xdc0000.0p-23, 0xc00000.0p-35 }, + { 0xdd0000.0p-23, 0xca0000.0p-33 }, + { 0xde0000.0p-23, -0xb80000.0p-34 }, + { 0xdf0000.0p-23, 0xd00000.0p-35 }, + { 0xe00000.0p-23, 0xc00000.0p-33 }, + { 0xe10000.0p-23, -0xf40000.0p-34 }, + { 0xe20000.0p-23, 0x800000.0p-37 }, + { 0xe30000.0p-23, 0x860000.0p-33 }, + { 0xe40000.0p-23, -0xc80000.0p-33 }, + { 0xe50000.0p-23, -0xa80000.0p-34 }, + { 0xe60000.0p-23, 0xe00000.0p-36 }, + { 0xe70000.0p-23, 0x880000.0p-33 }, + { 0xe80000.0p-23, -0xe00000.0p-33 }, + { 0xe90000.0p-23, -0xfc0000.0p-34 }, + { 0xea0000.0p-23, -0x800000.0p-35 }, + { 0xeb0000.0p-23, 0xe80000.0p-35 }, + { 0xec0000.0p-23, 0x900000.0p-33 }, + { 0xed0000.0p-23, 0xe20000.0p-33 }, + { 0xee0000.0p-23, -0xac0000.0p-33 }, + { 0xef0000.0p-23, -0xc80000.0p-34 }, + { 0xf00000.0p-23, -0x800000.0p-35 }, + { 0xf10000.0p-23, 0x800000.0p-35 }, + { 0xf20000.0p-23, 0xb80000.0p-34 }, + { 0xf30000.0p-23, 0x940000.0p-33 }, + { 0xf40000.0p-23, 0xc80000.0p-33 }, + { 0xf50000.0p-23, -0xf20000.0p-33 }, + { 0xf60000.0p-23, -0xc80000.0p-33 }, + { 0xf70000.0p-23, -0xa20000.0p-33 }, + { 0xf80000.0p-23, -0x800000.0p-33 }, + { 0xf90000.0p-23, -0xc40000.0p-34 }, + { 0xfa0000.0p-23, -0x900000.0p-34 }, + { 0xfb0000.0p-23, -0xc80000.0p-35 }, + { 0xfc0000.0p-23, -0x800000.0p-35 }, + { 0xfd0000.0p-23, -0x900000.0p-36 }, + { 0xfe0000.0p-23, -0x800000.0p-37 }, + { 0xff0000.0p-23, -0x800000.0p-39 }, + { 0x800000.0p-22, 0 }, }; #endif /* USE_UTAB */ #ifdef STRUCT_RETURN #define RETURN1(rp, v) do { \ (rp)->hi = (v); \ (rp)->lo_set = 0; \ return; \ } while (0) #define RETURN2(rp, h, l) do { \ (rp)->hi = (h); \ (rp)->lo = (l); \ (rp)->lo_set = 1; \ return; \ } while (0) struct ld { long double hi; long double lo; int lo_set; }; #else #define RETURN1(rp, v) RETURNF(v) #define RETURN2(rp, h, l) RETURNI((h) + (l)) #endif #ifdef STRUCT_RETURN static inline __always_inline void k_logl(long double x, struct ld *rp) #else long double logl(long double x) #endif { long double d, dk, val_hi, val_lo, z; uint64_t ix, lx; int i, k; uint16_t hx; EXTRACT_LDBL80_WORDS(hx, lx, x); k = -16383; #if 0 /* Hard to do efficiently. Don't do it until we support all modes. */ if (x == 1) RETURN1(rp, 0); /* log(1) = +0 in all rounding modes */ #endif if (hx == 0 || hx >= 0x8000) { /* zero, negative or subnormal? */ if (((hx & 0x7fff) | lx) == 0) RETURN1(rp, -1 / zero); /* log(+-0) = -Inf */ if (hx != 0) /* log(neg or [pseudo-]NaN) = qNaN: */ RETURN1(rp, (x - x) / zero); x *= 0x1.0p65; /* subnormal; scale up x */ /* including pseudo-subnormals */ EXTRACT_LDBL80_WORDS(hx, lx, x); k = -16383 - 65; } else if (hx >= 0x7fff || (lx & 0x8000000000000000ULL) == 0) RETURN1(rp, x + x); /* log(Inf or NaN) = Inf or qNaN */ /* log(pseudo-Inf) = qNaN */ /* log(pseudo-NaN) = qNaN */ /* log(unnormal) = qNaN */ #ifndef STRUCT_RETURN ENTERI(); #endif k += hx; ix = lx & 0x7fffffffffffffffULL; dk = k; /* Scale x to be in [1, 2). */ SET_LDBL_EXPSIGN(x, 0x3fff); /* 0 <= i <= INTERVALS: */ #define L2I (64 - LOG2_INTERVALS) i = (ix + (1LL << (L2I - 2))) >> (L2I - 1); /* * -0.005280 < d < 0.004838. In particular, the infinite- * precision |d| is <= 2**-7. Rounding of G(i) to 8 bits * ensures that d is representable without extra precision for * this bound on |d| (since when this calculation is expressed * as x*G(i)-1, the multiplication needs as many extra bits as * G(i) has and the subtraction cancels 8 bits). But for * most i (107 cases out of 129), the infinite-precision |d| * is <= 2**-8. G(i) is rounded to 9 bits for such i to give * better accuracy (this works by improving the bound on |d|, * which in turn allows rounding to 9 bits in more cases). * This is only important when the original x is near 1 -- it * lets us avoid using a special method to give the desired * accuracy for such x. */ if (0) d = x * G(i) - 1; else { #ifdef USE_UTAB d = (x - H(i)) * G(i) + E(i); #else long double x_hi, x_lo; float fx_hi; /* * Split x into x_hi + x_lo to calculate x*G(i)-1 exactly. * G(i) has at most 9 bits, so the splitting point is not * critical. */ SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000); x_hi = fx_hi; x_lo = x - x_hi; d = x_hi * G(i) - 1 + x_lo * G(i); #endif } /* * Our algorithm depends on exact cancellation of F_lo(i) and * F_hi(i) with dk*ln_2_lo and dk*ln2_hi when k is -1 and i is * at the end of the table. This and other technical complications * make it difficult to avoid the double scaling in (dk*ln2) * * log(base) for base != e without losing more accuracy and/or * efficiency than is gained. */ z = d * d; val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) + (F_lo(i) + dk * ln2_lo + z * d * (d * P4 + P3)) + z * P2; val_hi = d; #ifdef DEBUG if (fetestexcept(FE_UNDERFLOW)) breakpoint(); #endif _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); RETURN2(rp, val_hi, val_lo); } long double log1pl(long double x) { long double d, d_hi, d_lo, dk, f_lo, val_hi, val_lo, z; long double f_hi, twopminusk; uint64_t ix, lx; int i, k; int16_t ax, hx; DOPRINT_START(&x); EXTRACT_LDBL80_WORDS(hx, lx, x); if (hx < 0x3fff) { /* x < 1, or x neg NaN */ ax = hx & 0x7fff; if (ax >= 0x3fff) { /* x <= -1, or x neg NaN */ if (ax == 0x3fff && lx == 0x8000000000000000ULL) RETURNP(-1 / zero); /* log1p(-1) = -Inf */ /* log1p(x < 1, or x [pseudo-]NaN) = qNaN: */ RETURNP((x - x) / (x - x)); } if (ax <= 0x3fbe) { /* |x| < 2**-64 */ if ((int)x == 0) RETURNP(x); /* x with inexact if x != 0 */ } f_hi = 1; f_lo = x; } else if (hx >= 0x7fff) { /* x +Inf or non-neg NaN */ RETURNP(x + x); /* log1p(Inf or NaN) = Inf or qNaN */ /* log1p(pseudo-Inf) = qNaN */ /* log1p(pseudo-NaN) = qNaN */ /* log1p(unnormal) = qNaN */ } else if (hx < 0x407f) { /* 1 <= x < 2**128 */ f_hi = x; f_lo = 1; } else { /* 2**128 <= x < +Inf */ f_hi = x; f_lo = 0; /* avoid underflow of the P5 term */ } ENTERI(); x = f_hi + f_lo; f_lo = (f_hi - x) + f_lo; EXTRACT_LDBL80_WORDS(hx, lx, x); k = -16383; k += hx; ix = lx & 0x7fffffffffffffffULL; dk = k; SET_LDBL_EXPSIGN(x, 0x3fff); twopminusk = 1; SET_LDBL_EXPSIGN(twopminusk, 0x7ffe - (hx & 0x7fff)); f_lo *= twopminusk; i = (ix + (1LL << (L2I - 2))) >> (L2I - 1); /* * x*G(i)-1 (with a reduced x) can be represented exactly, as * above, but now we need to evaluate the polynomial on d = * (x+f_lo)*G(i)-1 and extra precision is needed for that. * Since x+x_lo is a hi+lo decomposition and subtracting 1 * doesn't lose too many bits, an inexact calculation for * f_lo*G(i) is good enough. */ if (0) d_hi = x * G(i) - 1; else { #ifdef USE_UTAB d_hi = (x - H(i)) * G(i) + E(i); #else long double x_hi, x_lo; float fx_hi; SET_FLOAT_WORD(fx_hi, (lx >> 40) | 0x3f800000); x_hi = fx_hi; x_lo = x - x_hi; d_hi = x_hi * G(i) - 1 + x_lo * G(i); #endif } d_lo = f_lo * G(i); /* * This is _2sumF(d_hi, d_lo) inlined. The condition * (d_hi == 0 || |d_hi| >= |d_lo|) for using _2sumF() is not * always satisifed, so it is not clear that this works, but * it works in practice. It works even if it gives a wrong * normalized d_lo, since |d_lo| > |d_hi| implies that i is * nonzero and d is tiny, so the F(i) term dominates d_lo. * In float precision: * (By exhaustive testing, the worst case is d_hi = 0x1.bp-25. * And if d is only a little tinier than that, we would have * another underflow problem for the P3 term; this is also ruled * out by exhaustive testing.) */ d = d_hi + d_lo; d_lo = d_hi - d + d_lo; d_hi = d; z = d * d; val_lo = z * d * z * (z * (d * P8 + P7) + (d * P6 + P5)) + (F_lo(i) + dk * ln2_lo + d_lo + z * d * (d * P4 + P3)) + z * P2; val_hi = d_hi; #ifdef DEBUG if (fetestexcept(FE_UNDERFLOW)) breakpoint(); #endif _3sumF(val_hi, val_lo, F_hi(i) + dk * ln2_hi); RETURN2PI(val_hi, val_lo); } #ifdef STRUCT_RETURN long double logl(long double x) { struct ld r; ENTERI(); DOPRINT_START(&x); k_logl(x, &r); RETURNSPI(&r); } static const double invln10_hi = 4.3429448190317999e-1, /* 0x1bcb7b1526e000.0p-54 */ invln10_lo = 7.1842412889749798e-14, /* 0x1438ca9aadd558.0p-96 */ invln2_hi = 1.4426950408887933e0, /* 0x171547652b8000.0p-52 */ invln2_lo = 1.7010652264631490e-13; /* 0x17f0bbbe87fed0.0p-95 */ long double log10l(long double x) { struct ld r; long double hi, lo; ENTERI(); DOPRINT_START(&x); k_logl(x, &r); if (!r.lo_set) RETURNPI(r.hi); _2sumF(r.hi, r.lo); hi = (float)r.hi; lo = r.lo + (r.hi - hi); RETURN2PI(invln10_hi * hi, (invln10_lo + invln10_hi) * lo + invln10_lo * hi); } long double log2l(long double x) { struct ld r; long double hi, lo; ENTERI(); DOPRINT_START(&x); k_logl(x, &r); if (!r.lo_set) RETURNPI(r.hi); _2sumF(r.hi, r.lo); hi = (float)r.hi; lo = r.lo + (r.hi - hi); RETURN2PI(invln2_hi * hi, (invln2_lo + invln2_hi) * lo + invln2_lo * hi); } #endif /* STRUCT_RETURN */