1 /*- 2 * Copyright (c) 2009-2012 Steven G. Kargl 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice unmodified, this list of conditions, and the following 10 * disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 25 * 26 * Optimized by Bruce D. Evans. 27 */ 28 29 #include <sys/cdefs.h> 30 __FBSDID("$FreeBSD$"); 31 32 /*- 33 * Compute the exponential of x for Intel 80-bit format. This is based on: 34 * 35 * PTP Tang, "Table-driven implementation of the exponential function 36 * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15, 37 * 144-157 (1989). 38 * 39 * where the 32 table entries have been expanded to INTERVALS (see below). 40 */ 41 42 #include <float.h> 43 44 #ifdef __i386__ 45 #include <ieeefp.h> 46 #endif 47 48 #include "fpmath.h" 49 #include "math.h" 50 #include "math_private.h" 51 52 #define INTERVALS 128 53 #define BIAS (LDBL_MAX_EXP - 1) 54 55 static const long double 56 huge = 0x1p10000L, 57 twom10000 = 0x1p-10000L; 58 /* XXX Prevent gcc from erroneously constant folding this: */ 59 static volatile const long double tiny = 0x1p-10000L; 60 61 static const union IEEEl2bits 62 /* log(2**16384 - 0.5) rounded towards zero: */ 63 o_threshold = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), 64 /* log(2**(-16381-64-1)) rounded towards zero: */ 65 u_threshold = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); 66 67 static const double 68 /* 69 * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must 70 * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest 71 * bits zero so that multiplication of it by n is exact. 72 */ 73 INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ 74 L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */ 75 L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */ 76 /* 77 * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]: 78 * |exp(x) - p(x)| < 2**-77.2 79 * (0.002708 is ln2/(2*INTERVALS) rounded up a little). 80 */ 81 P2 = 0.5, 82 P3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */ 83 P4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */ 84 P5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */ 85 P6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */ 86 87 /* 88 * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where 89 * the first 53 bits of the significand are stored in hi and the next 53 90 * bits are in lo. Tang's paper states that the trailing 6 bits of hi must 91 * be zero for his algorithm in both single and double precision, because 92 * the table is re-used in the implementation of expm1() where a floating 93 * point addition involving hi must be exact. Here hi is double, so 94 * converting it to long double gives 11 trailing zero bits. 95 */ 96 static const struct { 97 double hi; 98 double lo; 99 /* XXX should rename 's'. */ 100 } s[INTERVALS] = { 101 0x1p+0, 0x0p+0, 102 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54, 103 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53, 104 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53, 105 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55, 106 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53, 107 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57, 108 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54, 109 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54, 110 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54, 111 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59, 112 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53, 113 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53, 114 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53, 115 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53, 116 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55, 117 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53, 118 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53, 119 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55, 120 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53, 121 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54, 122 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53, 123 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55, 124 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55, 125 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54, 126 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55, 127 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55, 128 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53, 129 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55, 130 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53, 131 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54, 132 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56, 133 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55, 134 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55, 135 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54, 136 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53, 137 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53, 138 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53, 139 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53, 140 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53, 141 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55, 142 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53, 143 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53, 144 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53, 145 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59, 146 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54, 147 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56, 148 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54, 149 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56, 150 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54, 151 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53, 152 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53, 153 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53, 154 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53, 155 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54, 156 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55, 157 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54, 158 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60, 159 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54, 160 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53, 161 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53, 162 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53, 163 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53, 164 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57, 165 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53, 166 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53, 167 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53, 168 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53, 169 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53, 170 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53, 171 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53, 172 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54, 173 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53, 174 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54, 175 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56, 176 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53, 177 0x1.82589994cce12p+0, 0x1.159f115f56694p-53, 178 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53, 179 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53, 180 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54, 181 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54, 182 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53, 183 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55, 184 0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53, 185 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53, 186 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53, 187 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53, 188 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53, 189 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56, 190 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56, 191 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53, 192 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54, 193 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53, 194 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54, 195 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54, 196 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53, 197 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54, 198 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53, 199 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53, 200 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53, 201 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53, 202 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53, 203 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55, 204 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53, 205 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55, 206 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54, 207 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54, 208 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56, 209 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56, 210 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53, 211 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53, 212 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53, 213 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55, 214 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53, 215 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54, 216 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54, 217 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53, 218 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53, 219 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55, 220 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54, 221 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53, 222 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53, 223 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54, 224 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54, 225 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54, 226 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53, 227 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55, 228 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 229 }; 230 231 long double 232 expl(long double x) 233 { 234 union IEEEl2bits u, v; 235 long double fn, q, r, r1, r2, t, t23, t45, twopk, twopkp10000, z; 236 int k, n, n2; 237 uint16_t hx, ix; 238 239 /* Filter out exceptional cases. */ 240 u.e = x; 241 hx = u.xbits.expsign; 242 ix = hx & 0x7fff; 243 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 244 if (ix == BIAS + LDBL_MAX_EXP) { 245 if (hx & 0x8000 && u.xbits.man == 1ULL << 63) 246 return (0.0L); /* x is -Inf */ 247 return (x + x); /* x is +Inf, NaN or unsupported */ 248 } 249 if (x > o_threshold.e) 250 return (huge * huge); 251 if (x < u_threshold.e) 252 return (tiny * tiny); 253 } else if (ix < BIAS - 66) { /* |x| < 0x1p-66 */ 254 /* includes pseudo-denormals */ 255 if (huge + x > 1.0L) /* trigger inexact iff x != 0 */ 256 return (1.0L + x); 257 } 258 259 ENTERI(); 260 261 /* Reduce x to (k*ln2 + midpoint[n2] + r1 + r2). */ 262 /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 263 fn = x * INV_L + 0x1.8p63 - 0x1.8p63; 264 r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */ 265 #if defined(HAVE_EFFICIENT_IRINTL) 266 n = irintl(fn); 267 #elif defined(HAVE_EFFICIENT_IRINT) 268 n = irint(fn); 269 #else 270 n = (int)fn; 271 #endif 272 n2 = (unsigned)n % INTERVALS; 273 k = (n - n2) / INTERVALS; 274 r1 = x - fn * L1; 275 r2 = -fn * L2; 276 277 /* Prepare scale factors. */ 278 v.xbits.man = 1ULL << 63; 279 if (k >= LDBL_MIN_EXP) { 280 v.xbits.expsign = BIAS + k; 281 twopk = v.e; 282 } else { 283 v.xbits.expsign = BIAS + k + 10000; 284 twopkp10000 = v.e; 285 } 286 287 /* Evaluate expl(midpoint[n2] + r1 + r2) = s[n2] * expl(r1 + r2). */ 288 /* Here q = q(r), not q(r1), since r1 is lopped like L1. */ 289 t45 = r * P5 + P4; 290 z = r * r; 291 t23 = r * P3 + P2; 292 q = r2 + z * t23 + z * z * t45 + z * z * z * P6; 293 t = (long double)s[n2].lo + s[n2].hi; 294 t = s[n2].lo + t * (q + r1) + s[n2].hi; 295 296 /* Scale by 2**k. */ 297 if (k >= LDBL_MIN_EXP) { 298 if (k == LDBL_MAX_EXP) 299 RETURNI(t * 2.0L * 0x1p16383L); 300 RETURNI(t * twopk); 301 } else { 302 RETURNI(t * twopkp10000 * twom10000); 303 } 304 } 305