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 BIAS (LDBL_MAX_EXP - 1) 53 54 static const long double 55 huge = 0x1p10000L, 56 twom10000 = 0x1p-10000L; 57 /* XXX Prevent gcc from erroneously constant folding this: */ 58 static volatile const long double tiny = 0x1p-10000L; 59 60 static const union IEEEl2bits 61 /* log(2**16384 - 0.5) rounded towards zero: */ 62 o_threshold = LD80C(0xb17217f7d1cf79ab, 13, 0, 11356.5234062941439488L), 63 /* log(2**(-16381-64-1)) rounded towards zero: */ 64 u_threshold = LD80C(0xb21dfe7f09e2baa9, 13, 1, -11399.4985314888605581L); 65 66 static const double __aligned(64) 67 /* 68 * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must 69 * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest 70 * bits zero so that multiplication of it by n is exact. 71 */ 72 L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */ 73 L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */ 74 INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ 75 /* 76 * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]: 77 * |exp(x) - p(x)| < 2**-77.2 78 * (0.002708 is ln2/(2*INTERVALS) rounded up a little). 79 */ 80 P2 = 0.5, 81 P3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */ 82 P4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */ 83 P5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */ 84 P6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */ 85 86 /* 87 * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where 88 * the first 47 (?!) bits of the significand is stored in hi and the next 53 89 * bits are in lo. 90 */ 91 #define INTERVALS 128 92 93 static const struct { 94 double hi; 95 double lo; 96 } s[INTERVALS] __aligned(16) = { 97 0x1p+0, 0x0p+0, 98 0x1.0163da9fb330p+0, 0x1.ab6c25335719bp-47, 99 0x1.02c9a3e77804p+0, 0x1.07737be56527cp-47, 100 0x1.04315e86e7f8p+0, 0x1.2f5ce3e688369p-50, 101 0x1.059b0d315854p+0, 0x1.a1d73e2a475b4p-47, 102 0x1.0706b29ddf6cp+0, 0x1.dc6dc403a9d88p-48, 103 0x1.0874518759bcp+0, 0x1.01186be4bb285p-49, 104 0x1.09e3ecac6f38p+0, 0x1.a290f03062c27p-51, 105 0x1.0b5586cf9890p+0, 0x1.ec5317256e308p-49, 106 0x1.0cc922b7247cp+0, 0x1.ba03db82dc49fp-47, 107 0x1.0e3ec32d3d18p+0, 0x1.10103a1727c58p-47, 108 0x1.0fb66affed30p+0, 0x1.af232091dd8a1p-48, 109 0x1.11301d0125b4p+0, 0x1.0a4ebbf1aed93p-48, 110 0x1.12abdc06c31cp+0, 0x1.7f72575a649adp-49, 111 0x1.1429aaea92dcp+0, 0x1.fb34101943b26p-48, 112 0x1.15a98c8a58e4p+0, 0x1.12480d573dd56p-48, 113 0x1.172b83c7d514p+0, 0x1.d6e6fbe462876p-47, 114 0x1.18af9388c8dcp+0, 0x1.4dddfb85cd1e1p-47, 115 0x1.1a35beb6fcb4p+0, 0x1.a9e5b4c7b4969p-47, 116 0x1.1bbe084045ccp+0, 0x1.39ab1e72b4428p-48, 117 0x1.1d4873168b98p+0, 0x1.53c02dc0144c8p-47, 118 0x1.1ed5022fcd90p+0, 0x1.cb8819ff61122p-48, 119 0x1.2063b88628ccp+0, 0x1.63b8eeb029509p-48, 120 0x1.21f49917ddc8p+0, 0x1.62552fd29294cp-48, 121 0x1.2387a6e75620p+0, 0x1.c3360fd6d8e0bp-47, 122 0x1.251ce4fb2a60p+0, 0x1.f9ac155bef4f5p-47, 123 0x1.26b4565e27ccp+0, 0x1.d257a673281d4p-48, 124 0x1.284dfe1f5638p+0, 0x1.2d9e2b9e07941p-53, 125 0x1.29e9df51fdecp+0, 0x1.09612e8afad12p-47, 126 0x1.2b87fd0dad98p+0, 0x1.ffbbd48ca71f9p-49, 127 0x1.2d285a6e4030p+0, 0x1.680123aa6da0fp-49, 128 0x1.2ecafa93e2f4p+0, 0x1.611ca0f45d524p-48, 129 0x1.306fe0a31b70p+0, 0x1.52de8d5a46306p-48, 130 0x1.32170fc4cd80p+0, 0x1.89a9ce78e1804p-47, 131 0x1.33c08b26416cp+0, 0x1.fa64e43086cb3p-47, 132 0x1.356c55f929fcp+0, 0x1.864a311a3b1bap-47, 133 0x1.371a7373aa9cp+0, 0x1.54e28aa05e8a9p-49, 134 0x1.38cae6d05d84p+0, 0x1.2c2d4e586cdf7p-47, 135 0x1.3a7db34e59fcp+0, 0x1.b750de494cf05p-47, 136 0x1.3c32dc313a8cp+0, 0x1.242000f9145acp-47, 137 0x1.3dea64c12340p+0, 0x1.11ada0911f09fp-47, 138 0x1.3fa4504ac800p+0, 0x1.ba0bf701aa418p-48, 139 0x1.4160a21f72e0p+0, 0x1.4fc2192dc79eep-47, 140 0x1.431f5d950a88p+0, 0x1.6dc704439410dp-48, 141 0x1.44e086061890p+0, 0x1.68189b7a04ef8p-47, 142 0x1.46a41ed1d004p+0, 0x1.772512f45922ap-48, 143 0x1.486a2b5c13ccp+0, 0x1.013c1a3b69063p-48, 144 0x1.4a32af0d7d3cp+0, 0x1.e672d8bcf46f9p-48, 145 0x1.4bfdad5362a0p+0, 0x1.38ea1cbd7f621p-47, 146 0x1.4dcb299fddd0p+0, 0x1.ac766dde353c2p-49, 147 0x1.4f9b2769d2c8p+0, 0x1.35699ec5b4d50p-47, 148 0x1.516daa2cf664p+0, 0x1.c112f52c84d82p-52, 149 0x1.5342b569d4f8p+0, 0x1.df0a83c49d86ap-52, 150 0x1.551a4ca5d920p+0, 0x1.d8a5d8c40486ap-49, 151 0x1.56f4736b527cp+0, 0x1.a66ecb004764fp-48, 152 0x1.58d12d497c7cp+0, 0x1.e9295e15b9a1ep-47, 153 0x1.5ab07dd48540p+0, 0x1.4ac64980a8c8fp-47, 154 0x1.5c9268a59468p+0, 0x1.b80e258dc0b4cp-47, 155 0x1.5e76f15ad214p+0, 0x1.0dd37c9840733p-49, 156 0x1.605e1b976dc0p+0, 0x1.160edeb25490ep-49, 157 0x1.6247eb03a558p+0, 0x1.2c7c3e81bf4b7p-50, 158 0x1.6434634ccc30p+0, 0x1.fc76f8714c4eep-48, 159 0x1.662388255220p+0, 0x1.24893ecf14dc8p-47, 160 0x1.68155d44ca94p+0, 0x1.9840e2b913dd0p-47, 161 0x1.6a09e667f3bcp+0, 0x1.921165f626cddp-49, 162 0x1.6c012750bda8p+0, 0x1.f76bb54cc007ap-47, 163 0x1.6dfb23c651a0p+0, 0x1.779107165f0dep-47, 164 0x1.6ff7df951948p+0, 0x1.e7c3f0da79f11p-51, 165 0x1.71f75e8ec5f4p+0, 0x1.9ee91b8797785p-47, 166 0x1.73f9a48a5814p+0, 0x1.9deae4d273456p-47, 167 0x1.75feb564267cp+0, 0x1.17edd35467491p-49, 168 0x1.780694fde5d0p+0, 0x1.fb0cd7014042cp-47, 169 0x1.7a11473eb018p+0, 0x1.b5f54408fdb37p-50, 170 0x1.7c1ed0130c10p+0, 0x1.93e2499a22c9cp-47, 171 0x1.7e2f336cf4e4p+0, 0x1.1082e815d0abdp-47, 172 0x1.80427543e1a0p+0, 0x1.1b60de67649a3p-48, 173 0x1.82589994cce0p+0, 0x1.28acf88afab35p-48, 174 0x1.8471a4623c78p+0, 0x1.667297b5cbe32p-47, 175 0x1.868d99b4492cp+0, 0x1.640720ec85613p-47, 176 0x1.88ac7d98a668p+0, 0x1.966530bcdf2d5p-48, 177 0x1.8ace5422aa0cp+0, 0x1.b5ba7c55a192dp-48, 178 0x1.8cf3216b5448p+0, 0x1.7de55439a2c39p-49, 179 0x1.8f1ae9915770p+0, 0x1.b15cc13a2e397p-47, 180 0x1.9145b0b91ffcp+0, 0x1.622986d1a7daep-50, 181 0x1.93737b0cdc5cp+0, 0x1.27a280e1f92a0p-47, 182 0x1.95a44cbc8520p+0, 0x1.dd36906d2b420p-49, 183 0x1.97d829fde4e4p+0, 0x1.f173d241f23d1p-49, 184 0x1.9a0f170ca078p+0, 0x1.cdd1884dc6234p-47, 185 0x1.9c49182a3f08p+0, 0x1.01c7c46b071f3p-48, 186 0x1.9e86319e3230p+0, 0x1.18c12653c7326p-47, 187 0x1.a0c667b5de54p+0, 0x1.2594d6d45c656p-47, 188 0x1.a309bec4a2d0p+0, 0x1.9ac60b8fbb86dp-47, 189 0x1.a5503b23e254p+0, 0x1.c8b424491caf8p-48, 190 0x1.a799e1330b34p+0, 0x1.86f2dfb2b158fp-48, 191 0x1.a9e6b5579fd8p+0, 0x1.fa1f5921deffap-47, 192 0x1.ac36bbfd3f34p+0, 0x1.ce06dcb351893p-47, 193 0x1.ae89f995ad38p+0, 0x1.6af439a68bb99p-47, 194 0x1.b0e07298db64p+0, 0x1.2c8421566fe38p-47, 195 0x1.b33a2b84f15cp+0, 0x1.d7b5fe873decap-47, 196 0x1.b59728de5590p+0, 0x1.cc71c40888b24p-47, 197 0x1.b7f76f2fb5e4p+0, 0x1.baa9ec206ad4fp-50, 198 0x1.ba5b030a1064p+0, 0x1.30819678d5eb7p-49, 199 0x1.bcc1e904bc1cp+0, 0x1.2247ba0f45b3dp-48, 200 0x1.bf2c25bd71e0p+0, 0x1.10811ae04a31cp-49, 201 0x1.c199bdd85528p+0, 0x1.c2220cb12a092p-48, 202 0x1.c40ab5fffd04p+0, 0x1.d368a6fc1078cp-47, 203 0x1.c67f12e57d14p+0, 0x1.694426ffa41e5p-49, 204 0x1.c8f6d9406e78p+0, 0x1.a88d65e24402ep-47, 205 0x1.cb720dcef904p+0, 0x1.48a81e5e8f4a5p-47, 206 0x1.cdf0b555dc3cp+0, 0x1.ce227c4ac7d63p-47, 207 0x1.d072d4a07894p+0, 0x1.dc68791790d0bp-47, 208 0x1.d2f87080d89cp+0, 0x1.8c56f091cc4f5p-47, 209 0x1.d5818dcfba48p+0, 0x1.c976816bad9b8p-50, 210 0x1.d80e316c9838p+0, 0x1.7bb84f9d04880p-48, 211 0x1.da9e603db328p+0, 0x1.5c2300696db53p-50, 212 0x1.dd321f301b44p+0, 0x1.025b4aef1e032p-47, 213 0x1.dfc97337b9b4p+0, 0x1.eb968cac39ed3p-48, 214 0x1.e264614f5a10p+0, 0x1.45093b0fd0bd7p-47, 215 0x1.e502ee78b3fcp+0, 0x1.b139e8980a9cdp-47, 216 0x1.e7a51fbc74c8p+0, 0x1.a5aa4594191bcp-51, 217 0x1.ea4afa2a490cp+0, 0x1.9858f73a18f5ep-48, 218 0x1.ecf482d8e67cp+0, 0x1.846d81897dca5p-47, 219 0x1.efa1bee615a0p+0, 0x1.3bb8fe90d496dp-47, 220 0x1.f252b376bba8p+0, 0x1.74e8696fc3639p-48, 221 0x1.f50765b6e454p+0, 0x1.9d3e12dd8a18bp-54, 222 0x1.f7bfdad9cbe0p+0, 0x1.38913b4bfe72cp-48, 223 0x1.fa7c1819e90cp+0, 0x1.82e90a7e74b26p-48, 224 0x1.fd3c22b8f71cp+0, 0x1.884badd25995ep-47 225 }; 226 227 long double 228 expl(long double x) 229 { 230 union IEEEl2bits u, v; 231 long double fn, r, r1, r2, q, t, t23, t45, twopk, twopkp10000, z; 232 int k, n, n2; 233 uint16_t hx, ix; 234 235 /* Filter out exceptional cases. */ 236 u.e = x; 237 hx = u.xbits.expsign; 238 ix = hx & 0x7fff; 239 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 240 if (ix == BIAS + LDBL_MAX_EXP) { 241 if (hx & 0x8000 && u.xbits.man == 1ULL << 63) 242 return (0.0L); /* x is -Inf */ 243 return (x + x); /* x is +Inf, NaN or unsupported */ 244 } 245 if (x > o_threshold.e) 246 return (huge * huge); 247 if (x < u_threshold.e) 248 return (tiny * tiny); 249 } else if (ix <= BIAS - 34) { /* |x| < 0x1p-33 */ 250 /* includes pseudo-denormals */ 251 if (huge + x > 1.0L) /* trigger inexact iff x != 0 */ 252 return (1.0L + x); 253 } 254 255 ENTERI(); 256 257 /* Reduce x to (k*ln2 + midpoint[n2] + r1 + r2). */ 258 /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 259 fn = x * INV_L + 0x1.8p63 - 0x1.8p63; 260 r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */ 261 #if defined(HAVE_EFFICIENT_IRINTL) 262 n = irintl(fn); 263 #elif defined(HAVE_EFFICIENT_IRINT) 264 n = irint(fn); 265 #else 266 n = (int)fn; 267 #endif 268 n2 = (unsigned)n % INTERVALS; /* Tang's j. */ 269 k = (n - n2) / INTERVALS; 270 r1 = x - fn * L1; 271 r2 = -fn * L2; 272 273 /* Prepare scale factors. */ 274 v.xbits.man = 1ULL << 63; 275 if (k >= LDBL_MIN_EXP) { 276 v.xbits.expsign = BIAS + k; 277 twopk = v.e; 278 } else { 279 v.xbits.expsign = BIAS + k + 10000; 280 twopkp10000 = v.e; 281 } 282 283 /* Evaluate expl(midpoint[n2] + r1 + r2) = s[n2] * expl(r1 + r2). */ 284 /* Here q = q(r), not q(r1), since r1 is lopped like L1. */ 285 t45 = r * P5 + P4; 286 z = r * r; 287 t23 = r * P3 + P2; 288 q = r2 + z * t23 + z * z * t45 + z * z * z * P6; 289 t = (long double)s[n2].lo + s[n2].hi; 290 t = s[n2].lo + t * (q + r1) + s[n2].hi; 291 292 /* Scale by 2**k. */ 293 if (k >= LDBL_MIN_EXP) { 294 if (k == LDBL_MAX_EXP) 295 RETURNI(t * 2.0L * 0x1p16383L); 296 RETURNI(t * twopk); 297 } else { 298 RETURNI(t * twopkp10000 * twom10000); 299 } 300 } 301