1 /* from: FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl */ 2 3 /*- 4 * SPDX-License-Identifier: BSD-2-Clause 5 * 6 * Copyright (c) 2009-2013 Steven G. Kargl 7 * All rights reserved. 8 * 9 * Redistribution and use in source and binary forms, with or without 10 * modification, are permitted provided that the following conditions 11 * are met: 12 * 1. Redistributions of source code must retain the above copyright 13 * notice unmodified, this list of conditions, and the following 14 * disclaimer. 15 * 2. Redistributions in binary form must reproduce the above copyright 16 * notice, this list of conditions and the following disclaimer in the 17 * documentation and/or other materials provided with the distribution. 18 * 19 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 20 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 21 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 22 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 23 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 24 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 25 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 26 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 27 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 28 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 29 * 30 * Optimized by Bruce D. Evans. 31 */ 32 33 /* 34 * See s_expl.c for more comments about __k_expl(). 35 * 36 * See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments 37 * about the secondary kernels. 38 */ 39 40 #define INTERVALS 128 41 #define LOG2_INTERVALS 7 42 #define BIAS (LDBL_MAX_EXP - 1) 43 44 static const double 45 /* 46 * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must 47 * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest 48 * bits zero so that multiplication of it by n is exact. 49 */ 50 INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ 51 L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */ 52 L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */ 53 /* 54 * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]: 55 * |exp(x) - p(x)| < 2**-77.2 56 * (0.002708 is ln2/(2*INTERVALS) rounded up a little). 57 */ 58 A2 = 0.5, 59 A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */ 60 A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */ 61 A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */ 62 A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */ 63 64 /* 65 * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where 66 * the first 53 bits of the significand are stored in hi and the next 53 67 * bits are in lo. Tang's paper states that the trailing 6 bits of hi must 68 * be zero for his algorithm in both single and double precision, because 69 * the table is re-used in the implementation of expm1() where a floating 70 * point addition involving hi must be exact. Here hi is double, so 71 * converting it to long double gives 11 trailing zero bits. 72 */ 73 static const struct { 74 double hi; 75 double lo; 76 } tbl[INTERVALS] = { 77 { 0x1p+0, 0x0p+0 }, 78 /* 79 * XXX hi is rounded down, and the formatting is not quite normal. 80 * But I rather like both. The 0x1.*p format is good for 4N+1 81 * mantissa bits. Rounding down makes the lo terms positive, 82 * so that the columnar formatting can be simpler. 83 */ 84 { 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54 }, 85 { 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53 }, 86 { 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53 }, 87 { 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55 }, 88 { 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53 }, 89 { 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57 }, 90 { 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54 }, 91 { 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54 }, 92 { 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54 }, 93 { 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59 }, 94 { 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53 }, 95 { 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53 }, 96 { 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53 }, 97 { 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53 }, 98 { 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55 }, 99 { 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53 }, 100 { 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53 }, 101 { 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55 }, 102 { 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53 }, 103 { 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54 }, 104 { 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53 }, 105 { 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55 }, 106 { 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55 }, 107 { 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54 }, 108 { 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55 }, 109 { 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55 }, 110 { 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53 }, 111 { 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55 }, 112 { 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53 }, 113 { 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54 }, 114 { 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56 }, 115 { 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55 }, 116 { 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55 }, 117 { 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54 }, 118 { 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53 }, 119 { 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53 }, 120 { 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53 }, 121 { 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53 }, 122 { 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53 }, 123 { 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55 }, 124 { 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53 }, 125 { 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53 }, 126 { 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53 }, 127 { 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59 }, 128 { 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54 }, 129 { 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56 }, 130 { 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54 }, 131 { 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56 }, 132 { 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54 }, 133 { 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53 }, 134 { 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53 }, 135 { 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53 }, 136 { 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53 }, 137 { 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54 }, 138 { 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55 }, 139 { 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54 }, 140 { 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60 }, 141 { 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54 }, 142 { 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53 }, 143 { 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53 }, 144 { 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53 }, 145 { 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53 }, 146 { 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57 }, 147 { 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53 }, 148 { 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53 }, 149 { 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53 }, 150 { 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53 }, 151 { 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53 }, 152 { 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53 }, 153 { 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53 }, 154 { 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54 }, 155 { 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53 }, 156 { 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54 }, 157 { 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56 }, 158 { 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53 }, 159 { 0x1.82589994cce12p+0, 0x1.159f115f56694p-53 }, 160 { 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53 }, 161 { 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53 }, 162 { 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54 }, 163 { 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54 }, 164 { 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53 }, 165 { 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55 }, 166 { 0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53 }, 167 { 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53 }, 168 { 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53 }, 169 { 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53 }, 170 { 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53 }, 171 { 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56 }, 172 { 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56 }, 173 { 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53 }, 174 { 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54 }, 175 { 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53 }, 176 { 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54 }, 177 { 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54 }, 178 { 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53 }, 179 { 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54 }, 180 { 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53 }, 181 { 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53 }, 182 { 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53 }, 183 { 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53 }, 184 { 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53 }, 185 { 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55 }, 186 { 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53 }, 187 { 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55 }, 188 { 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54 }, 189 { 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54 }, 190 { 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56 }, 191 { 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56 }, 192 { 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53 }, 193 { 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53 }, 194 { 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53 }, 195 { 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55 }, 196 { 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53 }, 197 { 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54 }, 198 { 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54 }, 199 { 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53 }, 200 { 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53 }, 201 { 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55 }, 202 { 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54 }, 203 { 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53 }, 204 { 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53 }, 205 { 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54 }, 206 { 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54 }, 207 { 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54 }, 208 { 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53 }, 209 { 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55 }, 210 { 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 } 211 }; 212 213 /* 214 * Kernel for expl(x). x must be finite and not tiny or huge. 215 * "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN). 216 * "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2). 217 */ 218 static inline void 219 __k_expl(long double x, long double *hip, long double *lop, int *kp) 220 { 221 long double fn, q, r, r1, r2, t, z; 222 int n, n2; 223 224 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 225 fn = rnintl(x * INV_L); 226 r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */ 227 n = irint(fn); 228 n2 = (unsigned)n % INTERVALS; 229 /* Depend on the sign bit being propagated: */ 230 *kp = n >> LOG2_INTERVALS; 231 r1 = x - fn * L1; 232 r2 = fn * -L2; 233 234 /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ 235 z = r * r; 236 #if 0 237 q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; 238 #else 239 q = r2 + z * A2 + z * r * (A3 + r * A4 + z * (A5 + r * A6)); 240 #endif 241 t = (long double)tbl[n2].lo + tbl[n2].hi; 242 *hip = tbl[n2].hi; 243 *lop = tbl[n2].lo + t * (q + r1); 244 } 245 246 static inline void 247 k_hexpl(long double x, long double *hip, long double *lop) 248 { 249 float twopkm1; 250 int k; 251 252 __k_expl(x, hip, lop, &k); 253 SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23)); 254 *hip *= twopkm1; 255 *lop *= twopkm1; 256 } 257 258 static inline long double 259 hexpl(long double x) 260 { 261 long double hi, lo, twopkm2; 262 int k; 263 264 twopkm2 = 1; 265 __k_expl(x, &hi, &lo, &k); 266 SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2); 267 return (lo + hi) * 2 * twopkm2; 268 } 269 270 #ifdef _COMPLEX_H 271 /* 272 * See ../src/k_exp.c for details. 273 */ 274 static inline long double complex 275 __ldexp_cexpl(long double complex z, int expt) 276 { 277 long double c, exp_x, hi, lo, s; 278 long double x, y, scale1, scale2; 279 int half_expt, k; 280 281 x = creall(z); 282 y = cimagl(z); 283 __k_expl(x, &hi, &lo, &k); 284 285 exp_x = (lo + hi) * 0x1p16382L; 286 expt += k - 16382; 287 288 scale1 = 1; 289 half_expt = expt / 2; 290 SET_LDBL_EXPSIGN(scale1, BIAS + half_expt); 291 scale2 = 1; 292 SET_LDBL_EXPSIGN(scale2, BIAS + expt - half_expt); 293 294 sincosl(y, &s, &c); 295 return (CMPLXL(c * exp_x * scale1 * scale2, 296 s * exp_x * scale1 * scale2)); 297 } 298 #endif /* _COMPLEX_H */ 299