s_expl.c revision 251343
1238722Skargl/*- 2251315Skargl * Copyright (c) 2009-2013 Steven G. Kargl 3238722Skargl * All rights reserved. 4238722Skargl * 5238722Skargl * Redistribution and use in source and binary forms, with or without 6238722Skargl * modification, are permitted provided that the following conditions 7238722Skargl * are met: 8238722Skargl * 1. Redistributions of source code must retain the above copyright 9238722Skargl * notice unmodified, this list of conditions, and the following 10238722Skargl * disclaimer. 11238722Skargl * 2. Redistributions in binary form must reproduce the above copyright 12238722Skargl * notice, this list of conditions and the following disclaimer in the 13238722Skargl * documentation and/or other materials provided with the distribution. 14238722Skargl * 15238722Skargl * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 16238722Skargl * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 17238722Skargl * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 18238722Skargl * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 19238722Skargl * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 20238722Skargl * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 21238722Skargl * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 22238722Skargl * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 23238722Skargl * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 24238722Skargl * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 25238722Skargl * 26238722Skargl * Optimized by Bruce D. Evans. 27238722Skargl */ 28238722Skargl 29238722Skargl#include <sys/cdefs.h> 30238722Skargl__FBSDID("$FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl $"); 31238722Skargl 32251316Skargl/** 33238722Skargl * Compute the exponential of x for Intel 80-bit format. This is based on: 34238722Skargl * 35238722Skargl * PTP Tang, "Table-driven implementation of the exponential function 36238722Skargl * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15, 37238722Skargl * 144-157 (1989). 38238722Skargl * 39238784Skargl * where the 32 table entries have been expanded to INTERVALS (see below). 40238722Skargl */ 41238722Skargl 42238722Skargl#include <float.h> 43238722Skargl 44238722Skargl#ifdef __i386__ 45238722Skargl#include <ieeefp.h> 46238722Skargl#endif 47238722Skargl 48238783Skargl#include "fpmath.h" 49238722Skargl#include "math.h" 50238722Skargl#include "math_private.h" 51238722Skargl 52240861Skargl#define INTERVALS 128 53251327Skargl#define LOG2_INTERVALS 7 54238722Skargl#define BIAS (LDBL_MAX_EXP - 1) 55238722Skargl 56238722Skarglstatic const long double 57238722Skarglhuge = 0x1p10000L, 58238722Skargltwom10000 = 0x1p-10000L; 59238722Skargl/* XXX Prevent gcc from erroneously constant folding this: */ 60238722Skarglstatic volatile const long double tiny = 0x1p-10000L; 61238722Skargl 62238722Skarglstatic const union IEEEl2bits 63238722Skargl/* log(2**16384 - 0.5) rounded towards zero: */ 64251328Skargl/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ 65251328Skarglo_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), 66251328Skargl#define o_threshold (o_thresholdu.e) 67238722Skargl/* log(2**(-16381-64-1)) rounded towards zero: */ 68251328Skarglu_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); 69251328Skargl#define u_threshold (u_thresholdu.e) 70238722Skargl 71241516Skarglstatic const double 72238722Skargl/* 73238784Skargl * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must 74238784Skargl * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest 75238784Skargl * bits zero so that multiplication of it by n is exact. 76238722Skargl */ 77240864SkarglINV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ 78238722SkarglL1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */ 79238722SkarglL2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */ 80238722Skargl/* 81238722Skargl * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]: 82238722Skargl * |exp(x) - p(x)| < 2**-77.2 83238784Skargl * (0.002708 is ln2/(2*INTERVALS) rounded up a little). 84238722Skargl */ 85251321SkarglA2 = 0.5, 86251321SkarglA3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */ 87251321SkarglA4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */ 88251321SkarglA5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */ 89251321SkarglA6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */ 90238722Skargl 91238722Skargl/* 92238784Skargl * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where 93241516Skargl * the first 53 bits of the significand are stored in hi and the next 53 94241516Skargl * bits are in lo. Tang's paper states that the trailing 6 bits of hi must 95240861Skargl * be zero for his algorithm in both single and double precision, because 96240861Skargl * the table is re-used in the implementation of expm1() where a floating 97241516Skargl * point addition involving hi must be exact. Here hi is double, so 98241516Skargl * converting it to long double gives 11 trailing zero bits. 99238722Skargl */ 100238722Skarglstatic const struct { 101238722Skargl double hi; 102238722Skargl double lo; 103251321Skargl} tbl[INTERVALS] = { 104238722Skargl 0x1p+0, 0x0p+0, 105240861Skargl 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54, 106240861Skargl 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53, 107240861Skargl 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53, 108240861Skargl 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55, 109240861Skargl 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53, 110240861Skargl 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57, 111240861Skargl 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54, 112240861Skargl 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54, 113240861Skargl 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54, 114240861Skargl 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59, 115240861Skargl 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53, 116240861Skargl 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53, 117240861Skargl 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53, 118240861Skargl 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53, 119240861Skargl 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55, 120240861Skargl 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53, 121240861Skargl 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53, 122240861Skargl 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55, 123240861Skargl 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53, 124240861Skargl 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54, 125240861Skargl 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53, 126240861Skargl 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55, 127240861Skargl 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55, 128240861Skargl 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54, 129240861Skargl 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55, 130240861Skargl 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55, 131240861Skargl 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53, 132240861Skargl 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55, 133240861Skargl 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53, 134240861Skargl 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54, 135240861Skargl 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56, 136240861Skargl 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55, 137240861Skargl 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55, 138240861Skargl 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54, 139240861Skargl 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53, 140240861Skargl 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53, 141240861Skargl 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53, 142240861Skargl 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53, 143240861Skargl 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53, 144240861Skargl 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55, 145240861Skargl 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53, 146240861Skargl 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53, 147240861Skargl 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53, 148240861Skargl 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59, 149240861Skargl 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54, 150240861Skargl 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56, 151240861Skargl 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54, 152240861Skargl 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56, 153240861Skargl 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54, 154240861Skargl 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53, 155240861Skargl 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53, 156240861Skargl 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53, 157240861Skargl 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53, 158240861Skargl 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54, 159240861Skargl 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55, 160240861Skargl 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54, 161240861Skargl 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60, 162240861Skargl 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54, 163240861Skargl 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53, 164240861Skargl 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53, 165240861Skargl 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53, 166240861Skargl 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53, 167240861Skargl 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57, 168240861Skargl 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53, 169240861Skargl 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53, 170240861Skargl 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53, 171240861Skargl 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53, 172240861Skargl 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53, 173240861Skargl 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53, 174240861Skargl 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53, 175240861Skargl 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54, 176240861Skargl 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53, 177240861Skargl 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54, 178240861Skargl 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56, 179240861Skargl 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53, 180240861Skargl 0x1.82589994cce12p+0, 0x1.159f115f56694p-53, 181240861Skargl 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53, 182240861Skargl 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53, 183240861Skargl 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54, 184240861Skargl 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54, 185240861Skargl 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53, 186240861Skargl 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55, 187240861Skargl 0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53, 188240861Skargl 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53, 189240861Skargl 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53, 190240861Skargl 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53, 191240861Skargl 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53, 192240861Skargl 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56, 193240861Skargl 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56, 194240861Skargl 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53, 195240861Skargl 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54, 196240861Skargl 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53, 197240861Skargl 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54, 198240861Skargl 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54, 199240861Skargl 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53, 200240861Skargl 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54, 201240861Skargl 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53, 202240861Skargl 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53, 203240861Skargl 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53, 204240861Skargl 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53, 205240861Skargl 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53, 206240861Skargl 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55, 207240861Skargl 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53, 208240861Skargl 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55, 209240861Skargl 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54, 210240861Skargl 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54, 211240861Skargl 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56, 212240861Skargl 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56, 213240861Skargl 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53, 214240861Skargl 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53, 215240861Skargl 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53, 216240861Skargl 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55, 217240861Skargl 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53, 218240861Skargl 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54, 219240861Skargl 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54, 220240861Skargl 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53, 221240861Skargl 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53, 222240861Skargl 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55, 223240861Skargl 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54, 224240861Skargl 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53, 225240861Skargl 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53, 226240861Skargl 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54, 227240861Skargl 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54, 228240861Skargl 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54, 229240861Skargl 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53, 230240861Skargl 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55, 231240861Skargl 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 232238722Skargl}; 233238722Skargl 234238722Skargllong double 235238722Skarglexpl(long double x) 236238722Skargl{ 237238722Skargl union IEEEl2bits u, v; 238251334Skargl long double fn, q, r, r1, r2, t, twopk, twopkp10000; 239251334Skargl long double z; 240238722Skargl int k, n, n2; 241238722Skargl uint16_t hx, ix; 242238722Skargl 243238722Skargl /* Filter out exceptional cases. */ 244238722Skargl u.e = x; 245238722Skargl hx = u.xbits.expsign; 246238722Skargl ix = hx & 0x7fff; 247238722Skargl if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 248238722Skargl if (ix == BIAS + LDBL_MAX_EXP) { 249251335Skargl if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 250251335Skargl return (-1 / x); 251251335Skargl return (x + x); /* x is +Inf, +NaN or unsupported */ 252238722Skargl } 253251328Skargl if (x > o_threshold) 254238722Skargl return (huge * huge); 255251328Skargl if (x < u_threshold) 256238722Skargl return (tiny * tiny); 257251335Skargl } else if (ix < BIAS - 65) { /* |x| < 0x1p-65 (includes pseudos) */ 258251335Skargl return (1 + x); /* 1 with inexact iff x != 0 */ 259238722Skargl } 260238722Skargl 261238722Skargl ENTERI(); 262238722Skargl 263251330Skargl /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 264238722Skargl /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 265238722Skargl fn = x * INV_L + 0x1.8p63 - 0x1.8p63; 266238722Skargl r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */ 267238722Skargl#if defined(HAVE_EFFICIENT_IRINTL) 268251325Skargl n = irintl(fn); 269238722Skargl#elif defined(HAVE_EFFICIENT_IRINT) 270251325Skargl n = irint(fn); 271238722Skargl#else 272251325Skargl n = (int)fn; 273238722Skargl#endif 274240864Skargl n2 = (unsigned)n % INTERVALS; 275251327Skargl /* Depend on the sign bit being propagated: */ 276251327Skargl k = n >> LOG2_INTERVALS; 277238722Skargl r1 = x - fn * L1; 278251338Skargl r2 = fn * -L2; 279238722Skargl 280238722Skargl /* Prepare scale factors. */ 281251339Skargl v.e = 1; 282238722Skargl if (k >= LDBL_MIN_EXP) { 283238722Skargl v.xbits.expsign = BIAS + k; 284238722Skargl twopk = v.e; 285238722Skargl } else { 286238722Skargl v.xbits.expsign = BIAS + k + 10000; 287238722Skargl twopkp10000 = v.e; 288238722Skargl } 289238722Skargl 290251334Skargl /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ 291238722Skargl z = r * r; 292251334Skargl q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 293251321Skargl t = (long double)tbl[n2].lo + tbl[n2].hi; 294251321Skargl t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; 295238722Skargl 296238722Skargl /* Scale by 2**k. */ 297238722Skargl if (k >= LDBL_MIN_EXP) { 298238722Skargl if (k == LDBL_MAX_EXP) 299251339Skargl RETURNI(t * 2 * 0x1p16383L); 300238722Skargl RETURNI(t * twopk); 301238722Skargl } else { 302238722Skargl RETURNI(t * twopkp10000 * twom10000); 303238722Skargl } 304238722Skargl} 305251343Skargl 306251343Skargl/** 307251343Skargl * Compute expm1l(x) for Intel 80-bit format. This is based on: 308251343Skargl * 309251343Skargl * PTP Tang, "Table-driven implementation of the Expm1 function 310251343Skargl * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, 311251343Skargl * 211-222 (1992). 312251343Skargl */ 313251343Skargl 314251343Skargl/* 315251343Skargl * Our T1 and T2 are chosen to be approximately the points where method 316251343Skargl * A and method B have the same accuracy. Tang's T1 and T2 are the 317251343Skargl * points where method A's accuracy changes by a full bit. For Tang, 318251343Skargl * this drop in accuracy makes method A immediately less accurate than 319251343Skargl * method B, but our larger INTERVALS makes method A 2 bits more 320251343Skargl * accurate so it remains the most accurate method significantly 321251343Skargl * closer to the origin despite losing the full bit in our extended 322251343Skargl * range for it. 323251343Skargl */ 324251343Skarglstatic const double 325251343SkarglT1 = -0.1659, /* ~-30.625/128 * log(2) */ 326251343SkarglT2 = 0.1659; /* ~30.625/128 * log(2) */ 327251343Skargl 328251343Skargl/* 329251343Skargl * Domain [-0.1659, 0.1659], range ~[-1.2027e-22, 3.4417e-22]: 330251343Skargl * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.2 331251343Skargl */ 332251343Skarglstatic const union IEEEl2bits 333251343SkarglB3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), 334251343SkarglB4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); 335251343Skargl 336251343Skarglstatic const double 337251343SkarglB5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ 338251343SkarglB6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ 339251343SkarglB7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ 340251343SkarglB8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ 341251343SkarglB9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ 342251343SkarglB10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ 343251343SkarglB11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ 344251343SkarglB12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ 345251343Skargl 346251343Skargllong double 347251343Skarglexpm1l(long double x) 348251343Skargl{ 349251343Skargl union IEEEl2bits u, v; 350251343Skargl long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; 351251343Skargl long double x_lo, x2, z; 352251343Skargl long double x4; 353251343Skargl int k, n, n2; 354251343Skargl uint16_t hx, ix; 355251343Skargl 356251343Skargl /* Filter out exceptional cases. */ 357251343Skargl u.e = x; 358251343Skargl hx = u.xbits.expsign; 359251343Skargl ix = hx & 0x7fff; 360251343Skargl if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ 361251343Skargl if (ix == BIAS + LDBL_MAX_EXP) { 362251343Skargl if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ 363251343Skargl return (-1 / x - 1); 364251343Skargl return (x + x); /* x is +Inf, +NaN or unsupported */ 365251343Skargl } 366251343Skargl if (x > o_threshold) 367251343Skargl return (huge * huge); 368251343Skargl /* 369251343Skargl * expm1l() never underflows, but it must avoid 370251343Skargl * unrepresentable large negative exponents. We used a 371251343Skargl * much smaller threshold for large |x| above than in 372251343Skargl * expl() so as to handle not so large negative exponents 373251343Skargl * in the same way as large ones here. 374251343Skargl */ 375251343Skargl if (hx & 0x8000) /* x <= -64 */ 376251343Skargl return (tiny - 1); /* good for x < -65ln2 - eps */ 377251343Skargl } 378251343Skargl 379251343Skargl ENTERI(); 380251343Skargl 381251343Skargl if (T1 < x && x < T2) { 382251343Skargl if (ix < BIAS - 64) { /* |x| < 0x1p-64 (includes pseudos) */ 383251343Skargl /* x (rounded) with inexact if x != 0: */ 384251343Skargl RETURNI(x == 0 ? x : 385251343Skargl (0x1p100 * x + fabsl(x)) * 0x1p-100); 386251343Skargl } 387251343Skargl 388251343Skargl x2 = x * x; 389251343Skargl x4 = x2 * x2; 390251343Skargl q = x4 * (x2 * (x4 * 391251343Skargl /* 392251343Skargl * XXX the number of terms is no longer good for 393251343Skargl * pairwise grouping of all except B3, and the 394251343Skargl * grouping is no longer from highest down. 395251343Skargl */ 396251343Skargl (x2 * B12 + (x * B11 + B10)) + 397251343Skargl (x2 * (x * B9 + B8) + (x * B7 + B6))) + 398251343Skargl (x * B5 + B4.e)) + x2 * x * B3.e; 399251343Skargl 400251343Skargl x_hi = (float)x; 401251343Skargl x_lo = x - x_hi; 402251343Skargl hx2_hi = x_hi * x_hi / 2; 403251343Skargl hx2_lo = x_lo * (x + x_hi) / 2; 404251343Skargl if (ix >= BIAS - 7) 405251343Skargl RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi)); 406251343Skargl else 407251343Skargl RETURNI(hx2_lo + q + hx2_hi + x); 408251343Skargl } 409251343Skargl 410251343Skargl /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 411251343Skargl /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 412251343Skargl fn = x * INV_L + 0x1.8p63 - 0x1.8p63; 413251343Skargl#if defined(HAVE_EFFICIENT_IRINTL) 414251343Skargl n = irintl(fn); 415251343Skargl#elif defined(HAVE_EFFICIENT_IRINT) 416251343Skargl n = irint(fn); 417251343Skargl#else 418251343Skargl n = (int)fn; 419251343Skargl#endif 420251343Skargl n2 = (unsigned)n % INTERVALS; 421251343Skargl k = n >> LOG2_INTERVALS; 422251343Skargl r1 = x - fn * L1; 423251343Skargl r2 = fn * -L2; 424251343Skargl r = r1 + r2; 425251343Skargl 426251343Skargl /* Prepare scale factor. */ 427251343Skargl v.e = 1; 428251343Skargl v.xbits.expsign = BIAS + k; 429251343Skargl twopk = v.e; 430251343Skargl 431251343Skargl /* 432251343Skargl * Evaluate lower terms of 433251343Skargl * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 434251343Skargl */ 435251343Skargl z = r * r; 436251343Skargl q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 437251343Skargl 438251343Skargl t = (long double)tbl[n2].lo + tbl[n2].hi; 439251343Skargl 440251343Skargl if (k == 0) { 441251343Skargl t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + 442251343Skargl (tbl[n2].hi - 1); 443251343Skargl RETURNI(t); 444251343Skargl } 445251343Skargl if (k == -1) { 446251343Skargl t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + 447251343Skargl (tbl[n2].hi - 2); 448251343Skargl RETURNI(t / 2); 449251343Skargl } 450251343Skargl if (k < -7) { 451251343Skargl t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; 452251343Skargl RETURNI(t * twopk - 1); 453251343Skargl } 454251343Skargl if (k > 2 * LDBL_MANT_DIG - 1) { 455251343Skargl t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; 456251343Skargl if (k == LDBL_MAX_EXP) 457251343Skargl RETURNI(t * 2 * 0x1p16383L - 1); 458251343Skargl RETURNI(t * twopk - 1); 459251343Skargl } 460251343Skargl 461251343Skargl v.xbits.expsign = BIAS - k; 462251343Skargl twomk = v.e; 463251343Skargl 464251343Skargl if (k > LDBL_MANT_DIG - 1) 465251343Skargl t = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi; 466251343Skargl else 467251343Skargl t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk); 468251343Skargl RETURNI(t * twopk); 469251343Skargl} 470