k_expl.h revision 331722
165205Scg/* from: FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl */ 265205Scg 365205Scg/*- 465205Scg * Copyright (c) 2009-2013 Steven G. Kargl 565205Scg * All rights reserved. 665205Scg * 765205Scg * Redistribution and use in source and binary forms, with or without 865205Scg * modification, are permitted provided that the following conditions 965205Scg * are met: 1065205Scg * 1. Redistributions of source code must retain the above copyright 1165205Scg * notice unmodified, this list of conditions, and the following 1265205Scg * disclaimer. 1365205Scg * 2. Redistributions in binary form must reproduce the above copyright 1465205Scg * notice, this list of conditions and the following disclaimer in the 1565205Scg * documentation and/or other materials provided with the distribution. 1665205Scg * 1765205Scg * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 1865205Scg * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 1965205Scg * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 2065205Scg * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 2165205Scg * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 2265205Scg * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 2365205Scg * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 2465205Scg * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 2565205Scg * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 2665205Scg * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 2765205Scg * 2865205Scg * Optimized by Bruce D. Evans. 29119287Simp */ 30119287Simp 3165205Scg#include <sys/cdefs.h> 3282180Scg__FBSDID("$FreeBSD: stable/11/lib/msun/ld80/k_expl.h 331722 2018-03-29 02:50:57Z eadler $"); 3382180Scg 3465205Scg/* 3565205Scg * See s_expl.c for more comments about __k_expl(). 3665205Scg * 3765205Scg * See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments 3865205Scg * about the secondary kernels. 3965205Scg */ 4065205Scg 4165205Scg#define INTERVALS 128 4265205Scg#define LOG2_INTERVALS 7 4365205Scg#define BIAS (LDBL_MAX_EXP - 1) 4465205Scg 4565205Scgstatic const double 4665205Scg/* 4765205Scg * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must 4865205Scg * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest 4965205Scg * bits zero so that multiplication of it by n is exact. 5065205Scg */ 5165205ScgINV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ 5265205ScgL1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */ 5365205ScgL2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */ 5465205Scg/* 5565205Scg * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]: 5665205Scg * |exp(x) - p(x)| < 2**-77.2 5765205Scg * (0.002708 is ln2/(2*INTERVALS) rounded up a little). 5865205Scg */ 5965205ScgA2 = 0.5, 6065205ScgA3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */ 6165205ScgA4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */ 6265205ScgA5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */ 6365205ScgA6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */ 6465205Scg 6565205Scg/* 6665205Scg * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where 6765205Scg * the first 53 bits of the significand are stored in hi and the next 53 6865205Scg * bits are in lo. Tang's paper states that the trailing 6 bits of hi must 6965205Scg * be zero for his algorithm in both single and double precision, because 7065205Scg * the table is re-used in the implementation of expm1() where a floating 7165205Scg * point addition involving hi must be exact. Here hi is double, so 7265205Scg * converting it to long double gives 11 trailing zero bits. 7365205Scg */ 7465205Scgstatic const struct { 7565205Scg double hi; 7665205Scg double lo; 7765205Scg} tbl[INTERVALS] = { 7865205Scg { 0x1p+0, 0x0p+0 }, 7965205Scg /* 8065205Scg * XXX hi is rounded down, and the formatting is not quite normal. 8165205Scg * But I rather like both. The 0x1.*p format is good for 4N+1 8265205Scg * mantissa bits. Rounding down makes the lo terms positive, 8365205Scg * so that the columnar formatting can be simpler. 8465205Scg */ 8565205Scg { 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54 }, 8665205Scg { 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53 }, 8765205Scg { 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53 }, 8865205Scg { 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55 }, 8965205Scg { 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53 }, 9065205Scg { 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57 }, 9165205Scg { 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54 }, 9265205Scg { 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54 }, 9365205Scg { 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54 }, 9465205Scg { 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59 }, 9565205Scg { 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53 }, 9665205Scg { 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53 }, 9765205Scg { 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53 }, 9865205Scg { 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53 }, 9984658Scg { 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55 }, 10065205Scg { 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53 }, 10165205Scg { 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53 }, 10265205Scg { 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55 }, 10365205Scg { 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53 }, 10465205Scg { 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54 }, 10574763Scg { 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53 }, 10665205Scg { 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55 }, 10765205Scg { 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55 }, 10865205Scg { 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54 }, 10965205Scg { 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55 }, 11065205Scg { 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55 }, 11165205Scg { 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53 }, 11265644Scg { 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55 }, 11365205Scg { 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53 }, 11465205Scg { 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54 }, 11565205Scg { 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56 }, 11674763Scg { 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55 }, 11765205Scg { 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55 }, 11865205Scg { 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54 }, 11965205Scg { 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53 }, 12065205Scg { 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53 }, 12165205Scg { 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53 }, 12265205Scg { 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53 }, 12365205Scg { 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53 }, 124102620Ssobomax { 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55 }, 125102620Ssobomax { 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53 }, 126102620Ssobomax { 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53 }, 127102620Ssobomax { 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53 }, 12865205Scg { 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59 }, 12965205Scg { 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54 }, 13065205Scg { 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56 }, 13165205Scg { 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54 }, 132102620Ssobomax { 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56 }, 133102620Ssobomax { 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54 }, 134102620Ssobomax { 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53 }, 135102620Ssobomax { 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53 }, 13665205Scg { 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53 }, 137102620Ssobomax { 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53 }, 138102620Ssobomax { 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54 }, 139102620Ssobomax { 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55 }, 14065205Scg { 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54 }, 141102620Ssobomax { 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60 }, 142102620Ssobomax { 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54 }, 143102620Ssobomax { 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53 }, 14465205Scg { 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53 }, 14565205Scg { 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53 }, 14665205Scg { 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53 }, 14765205Scg { 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57 }, 14865205Scg { 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53 }, 14965205Scg { 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53 }, 15065205Scg { 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53 }, 15165205Scg { 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53 }, 15265205Scg { 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53 }, 15365205Scg { 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53 }, 15465205Scg { 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53 }, 15565205Scg { 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54 }, 15665205Scg { 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53 }, 15765205Scg { 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54 }, 15865205Scg { 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56 }, 15965205Scg { 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53 }, 16065205Scg { 0x1.82589994cce12p+0, 0x1.159f115f56694p-53 }, 161102620Ssobomax { 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53 }, 16284658Scg { 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53 }, 163102620Ssobomax { 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54 }, 164102889Ssobomax { 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54 }, 165102889Ssobomax { 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53 }, 16665205Scg { 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55 }, 16765205Scg { 0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53 }, 16865205Scg { 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53 }, 16965205Scg { 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53 }, 17065205Scg { 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53 }, 17165205Scg { 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53 }, 17265205Scg { 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56 }, 17365205Scg { 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56 }, 17465205Scg { 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53 }, 17565205Scg { 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54 }, 17665205Scg { 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53 }, 17765205Scg { 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54 }, 17865205Scg { 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54 }, 17965205Scg { 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53 }, 18065205Scg { 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54 }, 18165205Scg { 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53 }, 18265205Scg { 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53 }, 18365205Scg { 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53 }, 18465205Scg { 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53 }, 18565205Scg { 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53 }, 18665205Scg { 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55 }, 187108064Ssemenu { 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53 }, 18865205Scg { 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55 }, 18965205Scg { 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54 }, 190108064Ssemenu { 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54 }, 191108064Ssemenu { 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56 }, 19265205Scg { 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56 }, 193108064Ssemenu { 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53 }, 194108064Ssemenu { 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53 }, 19565205Scg { 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53 }, 196108064Ssemenu { 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55 }, 197108064Ssemenu { 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53 }, 19865205Scg { 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54 }, 19965205Scg { 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54 }, 20065205Scg { 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53 }, 20170134Scg { 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53 }, 20265205Scg { 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55 }, 20365205Scg { 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54 }, 20465205Scg { 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53 }, 20565205Scg { 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53 }, 20670134Scg { 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54 }, 20770134Scg { 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54 }, 20865205Scg { 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54 }, 20965205Scg { 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53 }, 21065205Scg { 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55 }, 21165340Scg { 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 } 21265205Scg}; 21365205Scg 21465205Scg/* 21565205Scg * Kernel for expl(x). x must be finite and not tiny or huge. 21665205Scg * "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN). 21765205Scg * "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2). 21865205Scg */ 21965205Scgstatic inline void 22065340Scg__k_expl(long double x, long double *hip, long double *lop, int *kp) 22165205Scg{ 22265205Scg long double fn, q, r, r1, r2, t, z; 22365205Scg int n, n2; 22465205Scg 22565205Scg /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 22665205Scg /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 22765205Scg fn = x * INV_L + 0x1.8p63 - 0x1.8p63; 22865205Scg r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */ 22965205Scg#if defined(HAVE_EFFICIENT_IRINTL) 230102889Ssobomax n = irintl(fn); 23165205Scg#elif defined(HAVE_EFFICIENT_IRINT) 23265340Scg n = irint(fn); 23365205Scg#else 23465205Scg n = (int)fn; 23565205Scg#endif 23670134Scg n2 = (unsigned)n % INTERVALS; 23770134Scg /* Depend on the sign bit being propagated: */ 23865205Scg *kp = n >> LOG2_INTERVALS; 23965205Scg r1 = x - fn * L1; 24065205Scg r2 = fn * -L2; 24165340Scg 24265205Scg /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ 24365340Scg z = r * r; 24465205Scg#if 0 24565340Scg q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 24665205Scg#else 24765205Scg q = r2 + z * A2 + z * r * (A3 + r * A4 + z * (A5 + r * A6)); 24865205Scg#endif 24965205Scg t = (long double)tbl[n2].lo + tbl[n2].hi; 25065205Scg *hip = tbl[n2].hi; 25165205Scg *lop = tbl[n2].lo + t * (q + r1); 25265205Scg} 25370134Scg 25465205Scgstatic inline void 25565340Scgk_hexpl(long double x, long double *hip, long double *lop) 25665205Scg{ 25765205Scg float twopkm1; 25865340Scg int k; 25965205Scg 26065205Scg __k_expl(x, hip, lop, &k); 26165205Scg SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23)); 26265205Scg *hip *= twopkm1; 26365205Scg *lop *= twopkm1; 26465205Scg} 26565205Scg 26670134Scgstatic inline long double 26765205Scghexpl(long double x) 26865205Scg{ 26970134Scg long double hi, lo, twopkm2; 27065205Scg int k; 27165205Scg 27270134Scg twopkm2 = 1; 27370134Scg __k_expl(x, &hi, &lo, &k); 27470134Scg SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2); 27570134Scg return (lo + hi) * 2 * twopkm2; 27670134Scg} 27770134Scg 27870134Scg#ifdef _COMPLEX_H 27970134Scg/* 28070134Scg * See ../src/k_exp.c for details. 28165340Scg */ 28265340Scgstatic inline long double complex 28365205Scg__ldexp_cexpl(long double complex z, int expt) 28465205Scg{ 28565205Scg long double exp_x, hi, lo; 28665205Scg long double x, y, scale1, scale2; 28765205Scg int half_expt, k; 28865205Scg 28965340Scg x = creall(z); 29065205Scg y = cimagl(z); 29165340Scg __k_expl(x, &hi, &lo, &k); 29265205Scg 29365205Scg exp_x = (lo + hi) * 0x1p16382; 29465205Scg expt += k - 16382; 29565205Scg 29665205Scg scale1 = 1; 29765205Scg half_expt = expt / 2; 29865205Scg SET_LDBL_EXPSIGN(scale1, BIAS + half_expt); 29965205Scg scale2 = 1; 30065340Scg SET_LDBL_EXPSIGN(scale1, BIAS + expt - half_expt); 30165205Scg 30265205Scg return (CMPLXL(cos(y) * exp_x * scale1 * scale2, 30365205Scg sinl(y) * exp_x * scale1 * scale2)); 30465205Scg} 30565205Scg#endif /* _COMPLEX_H */ 30665205Scg