1/* origin: FreeBSD /usr/src/lib/msun/ld80/s_exp2l.c and /usr/src/lib/msun/ld128/s_exp2l.c */ 2/*- 3 * Copyright (c) 2005-2008 David Schultz <das@FreeBSD.ORG> 4 * All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following 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 AND CONTRIBUTORS ``AS IS'' AND 16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 25 * SUCH DAMAGE. 26 */ 27 28#include "libm.h" 29 30#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 31long double exp2l(long double x) 32{ 33 return exp2(x); 34} 35#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 36#define TBLBITS 7 37#define TBLSIZE (1 << TBLBITS) 38 39static const double 40redux = 0x1.8p63 / TBLSIZE, 41P1 = 0x1.62e42fefa39efp-1, 42P2 = 0x1.ebfbdff82c58fp-3, 43P3 = 0x1.c6b08d7049fap-5, 44P4 = 0x1.3b2ab6fba4da5p-7, 45P5 = 0x1.5d8804780a736p-10, 46P6 = 0x1.430918835e33dp-13; 47 48static const double tbl[TBLSIZE * 2] = { 49 0x1.6a09e667f3bcdp-1, -0x1.bdd3413b2648p-55, 50 0x1.6c012750bdabfp-1, -0x1.2895667ff0cp-57, 51 0x1.6dfb23c651a2fp-1, -0x1.bbe3a683c88p-58, 52 0x1.6ff7df9519484p-1, -0x1.83c0f25860fp-56, 53 0x1.71f75e8ec5f74p-1, -0x1.16e4786887bp-56, 54 0x1.73f9a48a58174p-1, -0x1.0a8d96c65d5p-55, 55 0x1.75feb564267c9p-1, -0x1.0245957316ep-55, 56 0x1.780694fde5d3fp-1, 0x1.866b80a0216p-55, 57 0x1.7a11473eb0187p-1, -0x1.41577ee0499p-56, 58 0x1.7c1ed0130c132p-1, 0x1.f124cd1164ep-55, 59 0x1.7e2f336cf4e62p-1, 0x1.05d02ba157ap-57, 60 0x1.80427543e1a12p-1, -0x1.27c86626d97p-55, 61 0x1.82589994cce13p-1, -0x1.d4c1dd41533p-55, 62 0x1.8471a4623c7adp-1, -0x1.8d684a341cep-56, 63 0x1.868d99b4492edp-1, -0x1.fc6f89bd4f68p-55, 64 0x1.88ac7d98a6699p-1, 0x1.994c2f37cb5p-55, 65 0x1.8ace5422aa0dbp-1, 0x1.6e9f156864bp-55, 66 0x1.8cf3216b5448cp-1, -0x1.0d55e32e9e4p-57, 67 0x1.8f1ae99157736p-1, 0x1.5cc13a2e397p-56, 68 0x1.9145b0b91ffc6p-1, -0x1.dd6792e5825p-55, 69 0x1.93737b0cdc5e5p-1, -0x1.75fc781b58p-58, 70 0x1.95a44cbc8520fp-1, -0x1.64b7c96a5fp-57, 71 0x1.97d829fde4e5p-1, -0x1.d185b7c1b86p-55, 72 0x1.9a0f170ca07bap-1, -0x1.173bd91cee6p-55, 73 0x1.9c49182a3f09p-1, 0x1.c7c46b071f2p-57, 74 0x1.9e86319e32323p-1, 0x1.824ca78e64cp-57, 75 0x1.a0c667b5de565p-1, -0x1.359495d1cd5p-55, 76 0x1.a309bec4a2d33p-1, 0x1.6305c7ddc368p-55, 77 0x1.a5503b23e255dp-1, -0x1.d2f6edb8d42p-55, 78 0x1.a799e1330b358p-1, 0x1.bcb7ecac564p-55, 79 0x1.a9e6b5579fdbfp-1, 0x1.0fac90ef7fdp-55, 80 0x1.ac36bbfd3f37ap-1, -0x1.f9234cae76dp-56, 81 0x1.ae89f995ad3adp-1, 0x1.7a1cd345dcc8p-55, 82 0x1.b0e07298db666p-1, -0x1.bdef54c80e4p-55, 83 0x1.b33a2b84f15fbp-1, -0x1.2805e3084d8p-58, 84 0x1.b59728de5593ap-1, -0x1.c71dfbbba6ep-55, 85 0x1.b7f76f2fb5e47p-1, -0x1.5584f7e54acp-57, 86 0x1.ba5b030a1064ap-1, -0x1.efcd30e5429p-55, 87 0x1.bcc1e904bc1d2p-1, 0x1.23dd07a2d9fp-56, 88 0x1.bf2c25bd71e09p-1, -0x1.efdca3f6b9c8p-55, 89 0x1.c199bdd85529cp-1, 0x1.11065895049p-56, 90 0x1.c40ab5fffd07ap-1, 0x1.b4537e083c6p-55, 91 0x1.c67f12e57d14bp-1, 0x1.2884dff483c8p-55, 92 0x1.c8f6d9406e7b5p-1, 0x1.1acbc48805cp-57, 93 0x1.cb720dcef9069p-1, 0x1.503cbd1e94ap-57, 94 0x1.cdf0b555dc3fap-1, -0x1.dd83b53829dp-56, 95 0x1.d072d4a07897cp-1, -0x1.cbc3743797a8p-55, 96 0x1.d2f87080d89f2p-1, -0x1.d487b719d858p-55, 97 0x1.d5818dcfba487p-1, 0x1.2ed02d75b37p-56, 98 0x1.d80e316c98398p-1, -0x1.11ec18bedep-55, 99 0x1.da9e603db3285p-1, 0x1.c2300696db5p-55, 100 0x1.dd321f301b46p-1, 0x1.2da5778f019p-55, 101 0x1.dfc97337b9b5fp-1, -0x1.1a5cd4f184b8p-55, 102 0x1.e264614f5a129p-1, -0x1.7b627817a148p-55, 103 0x1.e502ee78b3ff6p-1, 0x1.39e8980a9cdp-56, 104 0x1.e7a51fbc74c83p-1, 0x1.2d522ca0c8ep-55, 105 0x1.ea4afa2a490dap-1, -0x1.e9c23179c288p-55, 106 0x1.ecf482d8e67f1p-1, -0x1.c93f3b411ad8p-55, 107 0x1.efa1bee615a27p-1, 0x1.dc7f486a4b68p-55, 108 0x1.f252b376bba97p-1, 0x1.3a1a5bf0d8e8p-55, 109 0x1.f50765b6e454p-1, 0x1.9d3e12dd8a18p-55, 110 0x1.f7bfdad9cbe14p-1, -0x1.dbb12d00635p-55, 111 0x1.fa7c1819e90d8p-1, 0x1.74853f3a593p-56, 112 0x1.fd3c22b8f71f1p-1, 0x1.2eb74966578p-58, 113 0x1p+0, 0x0p+0, 114 0x1.0163da9fb3335p+0, 0x1.b61299ab8cd8p-54, 115 0x1.02c9a3e778061p+0, -0x1.19083535b08p-56, 116 0x1.04315e86e7f85p+0, -0x1.0a31c1977c98p-54, 117 0x1.059b0d3158574p+0, 0x1.d73e2a475b4p-55, 118 0x1.0706b29ddf6dep+0, -0x1.c91dfe2b13cp-55, 119 0x1.0874518759bc8p+0, 0x1.186be4bb284p-57, 120 0x1.09e3ecac6f383p+0, 0x1.14878183161p-54, 121 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc61p-54, 122 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f8p-54, 123 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c58p-59, 124 0x1.0fb66affed31bp+0, -0x1.b9bedc44ebcp-57, 125 0x1.11301d0125b51p+0, -0x1.6c51039449bp-54, 126 0x1.12abdc06c31ccp+0, -0x1.1b514b36ca8p-58, 127 0x1.1429aaea92dep+0, -0x1.32fbf9af1368p-54, 128 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeabp-55, 129 0x1.172b83c7d517bp+0, -0x1.19041b9d78ap-55, 130 0x1.18af9388c8deap+0, -0x1.11023d1970f8p-54, 131 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4969p-55, 132 0x1.1bbe084045cd4p+0, -0x1.95386352ef6p-54, 133 0x1.1d4873168b9aap+0, 0x1.e016e00a264p-54, 134 0x1.1ed5022fcd91dp+0, -0x1.1df98027bb78p-54, 135 0x1.2063b88628cd6p+0, 0x1.dc775814a85p-55, 136 0x1.21f49917ddc96p+0, 0x1.2a97e9494a6p-55, 137 0x1.2387a6e756238p+0, 0x1.9b07eb6c7058p-54, 138 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f5p-55, 139 0x1.26b4565e27cddp+0, 0x1.2bd339940eap-55, 140 0x1.284dfe1f56381p+0, -0x1.a4c3a8c3f0d8p-54, 141 0x1.29e9df51fdee1p+0, 0x1.612e8afad12p-55, 142 0x1.2b87fd0dad99p+0, -0x1.10adcd6382p-59, 143 0x1.2d285a6e4030bp+0, 0x1.0024754db42p-54, 144 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d524p-56, 145 0x1.306fe0a31b715p+0, 0x1.6f46ad23183p-55, 146 0x1.32170fc4cd831p+0, 0x1.a9ce78e1804p-55, 147 0x1.33c08b26416ffp+0, 0x1.327218436598p-54, 148 0x1.356c55f929ff1p+0, -0x1.b5cee5c4e46p-55, 149 0x1.371a7373aa9cbp+0, -0x1.63aeabf42ebp-54, 150 0x1.38cae6d05d866p+0, -0x1.e958d3c99048p-54, 151 0x1.3a7db34e59ff7p+0, -0x1.5e436d661f6p-56, 152 0x1.3c32dc313a8e5p+0, -0x1.efff8375d2ap-54, 153 0x1.3dea64c123422p+0, 0x1.ada0911f09fp-55, 154 0x1.3fa4504ac801cp+0, -0x1.7d023f956fap-54, 155 0x1.4160a21f72e2ap+0, -0x1.ef3691c309p-58, 156 0x1.431f5d950a897p+0, -0x1.1c7dde35f7ap-55, 157 0x1.44e086061892dp+0, 0x1.89b7a04ef8p-59, 158 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a8p-54, 159 0x1.486a2b5c13cdp+0, 0x1.3c1a3b69062p-56, 160 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be8p-54, 161 0x1.4bfdad5362a27p+0, 0x1.d4397afec42p-56, 162 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a78p-54, 163 0x1.4f9b2769d2ca7p+0, -0x1.4b309d25958p-54, 164 0x1.516daa2cf6642p+0, -0x1.f768569bd94p-55, 165 0x1.5342b569d4f82p+0, -0x1.07abe1db13dp-55, 166 0x1.551a4ca5d920fp+0, -0x1.d689cefede6p-55, 167 0x1.56f4736b527dap+0, 0x1.9bb2c011d938p-54, 168 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1ep-55, 169 0x1.5ab07dd485429p+0, 0x1.6324c0546478p-54, 170 0x1.5c9268a5946b7p+0, 0x1.c4b1b81698p-60, 171 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e68p-54, 172 0x1.605e1b976dc09p+0, -0x1.3e2429b56de8p-54, 173 0x1.6247eb03a5585p+0, -0x1.383c17e40b48p-54, 174 0x1.6434634ccc32p+0, -0x1.c483c759d89p-55, 175 0x1.6623882552225p+0, -0x1.bb60987591cp-54, 176 0x1.68155d44ca973p+0, 0x1.038ae44f74p-57, 177}; 178 179/* 180 * exp2l(x): compute the base 2 exponential of x 181 * 182 * Accuracy: Peak error < 0.511 ulp. 183 * 184 * Method: (equally-spaced tables) 185 * 186 * Reduce x: 187 * x = 2**k + y, for integer k and |y| <= 1/2. 188 * Thus we have exp2l(x) = 2**k * exp2(y). 189 * 190 * Reduce y: 191 * y = i/TBLSIZE + z for integer i near y * TBLSIZE. 192 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), 193 * with |z| <= 2**-(TBLBITS+1). 194 * 195 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a 196 * degree-6 minimax polynomial with maximum error under 2**-69. 197 * The table entries each have 104 bits of accuracy, encoded as 198 * a pair of double precision values. 199 */ 200long double exp2l(long double x) 201{ 202 union ldshape u = {x}; 203 int e = u.i.se & 0x7fff; 204 long double r, z; 205 uint32_t i0; 206 union {uint32_t u; int32_t i;} k; 207 208 /* Filter out exceptional cases. */ 209 if (e >= 0x3fff + 13) { /* |x| >= 8192 or x is NaN */ 210 if (u.i.se >= 0x3fff + 14 && u.i.se >> 15 == 0) 211 /* overflow */ 212 return x * 0x1p16383L; 213 if (e == 0x7fff) /* -inf or -nan */ 214 return -1/x; 215 if (x < -16382) { 216 if (x <= -16446 || x - 0x1p63 + 0x1p63 != x) 217 /* underflow */ 218 FORCE_EVAL((float)(-0x1p-149/x)); 219 if (x <= -16446) 220 return 0; 221 } 222 } else if (e < 0x3fff - 64) { 223 return 1 + x; 224 } 225 226 /* 227 * Reduce x, computing z, i0, and k. The low bits of x + redux 228 * contain the 16-bit integer part of the exponent (k) followed by 229 * TBLBITS fractional bits (i0). We use bit tricks to extract these 230 * as integers, then set z to the remainder. 231 * 232 * Example: Suppose x is 0xabc.123456p0 and TBLBITS is 8. 233 * Then the low-order word of x + redux is 0x000abc12, 234 * We split this into k = 0xabc and i0 = 0x12 (adjusted to 235 * index into the table), then we compute z = 0x0.003456p0. 236 */ 237 u.f = x + redux; 238 i0 = u.i.m + TBLSIZE / 2; 239 k.u = i0 / TBLSIZE * TBLSIZE; 240 k.i /= TBLSIZE; 241 i0 %= TBLSIZE; 242 u.f -= redux; 243 z = x - u.f; 244 245 /* Compute r = exp2l(y) = exp2lt[i0] * p(z). */ 246 long double t_hi = tbl[2*i0]; 247 long double t_lo = tbl[2*i0 + 1]; 248 /* XXX This gives > 1 ulp errors outside of FE_TONEAREST mode */ 249 r = t_lo + (t_hi + t_lo) * z * (P1 + z * (P2 + z * (P3 + z * (P4 250 + z * (P5 + z * P6))))) + t_hi; 251 252 return scalbnl(r, k.i); 253} 254#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 255#define TBLBITS 7 256#define TBLSIZE (1 << TBLBITS) 257 258static const long double 259 P1 = 0x1.62e42fefa39ef35793c7673007e6p-1L, 260 P2 = 0x1.ebfbdff82c58ea86f16b06ec9736p-3L, 261 P3 = 0x1.c6b08d704a0bf8b33a762bad3459p-5L, 262 P4 = 0x1.3b2ab6fba4e7729ccbbe0b4f3fc2p-7L, 263 P5 = 0x1.5d87fe78a67311071dee13fd11d9p-10L, 264 P6 = 0x1.430912f86c7876f4b663b23c5fe5p-13L; 265 266static const double 267 P7 = 0x1.ffcbfc588b041p-17, 268 P8 = 0x1.62c0223a5c7c7p-20, 269 P9 = 0x1.b52541ff59713p-24, 270 P10 = 0x1.e4cf56a391e22p-28, 271 redux = 0x1.8p112 / TBLSIZE; 272 273static const long double tbl[TBLSIZE] = { 274 0x1.6a09e667f3bcc908b2fb1366dfeap-1L, 275 0x1.6c012750bdabeed76a99800f4edep-1L, 276 0x1.6dfb23c651a2ef220e2cbe1bc0d4p-1L, 277 0x1.6ff7df9519483cf87e1b4f3e1e98p-1L, 278 0x1.71f75e8ec5f73dd2370f2ef0b148p-1L, 279 0x1.73f9a48a58173bd5c9a4e68ab074p-1L, 280 0x1.75feb564267c8bf6e9aa33a489a8p-1L, 281 0x1.780694fde5d3f619ae02808592a4p-1L, 282 0x1.7a11473eb0186d7d51023f6ccb1ap-1L, 283 0x1.7c1ed0130c1327c49334459378dep-1L, 284 0x1.7e2f336cf4e62105d02ba1579756p-1L, 285 0x1.80427543e1a11b60de67649a3842p-1L, 286 0x1.82589994cce128acf88afab34928p-1L, 287 0x1.8471a4623c7acce52f6b97c6444cp-1L, 288 0x1.868d99b4492ec80e41d90ac2556ap-1L, 289 0x1.88ac7d98a669966530bcdf2d4cc0p-1L, 290 0x1.8ace5422aa0db5ba7c55a192c648p-1L, 291 0x1.8cf3216b5448bef2aa1cd161c57ap-1L, 292 0x1.8f1ae991577362b982745c72eddap-1L, 293 0x1.9145b0b91ffc588a61b469f6b6a0p-1L, 294 0x1.93737b0cdc5e4f4501c3f2540ae8p-1L, 295 0x1.95a44cbc8520ee9b483695a0e7fep-1L, 296 0x1.97d829fde4e4f8b9e920f91e8eb6p-1L, 297 0x1.9a0f170ca07b9ba3109b8c467844p-1L, 298 0x1.9c49182a3f0901c7c46b071f28dep-1L, 299 0x1.9e86319e323231824ca78e64c462p-1L, 300 0x1.a0c667b5de564b29ada8b8cabbacp-1L, 301 0x1.a309bec4a2d3358c171f770db1f4p-1L, 302 0x1.a5503b23e255c8b424491caf88ccp-1L, 303 0x1.a799e1330b3586f2dfb2b158f31ep-1L, 304 0x1.a9e6b5579fdbf43eb243bdff53a2p-1L, 305 0x1.ac36bbfd3f379c0db966a3126988p-1L, 306 0x1.ae89f995ad3ad5e8734d17731c80p-1L, 307 0x1.b0e07298db66590842acdfc6fb4ep-1L, 308 0x1.b33a2b84f15faf6bfd0e7bd941b0p-1L, 309 0x1.b59728de559398e3881111648738p-1L, 310 0x1.b7f76f2fb5e46eaa7b081ab53ff6p-1L, 311 0x1.ba5b030a10649840cb3c6af5b74cp-1L, 312 0x1.bcc1e904bc1d2247ba0f45b3d06cp-1L, 313 0x1.bf2c25bd71e088408d7025190cd0p-1L, 314 0x1.c199bdd85529c2220cb12a0916bap-1L, 315 0x1.c40ab5fffd07a6d14df820f17deap-1L, 316 0x1.c67f12e57d14b4a2137fd20f2a26p-1L, 317 0x1.c8f6d9406e7b511acbc48805c3f6p-1L, 318 0x1.cb720dcef90691503cbd1e949d0ap-1L, 319 0x1.cdf0b555dc3f9c44f8958fac4f12p-1L, 320 0x1.d072d4a07897b8d0f22f21a13792p-1L, 321 0x1.d2f87080d89f18ade123989ea50ep-1L, 322 0x1.d5818dcfba48725da05aeb66dff8p-1L, 323 0x1.d80e316c98397bb84f9d048807a0p-1L, 324 0x1.da9e603db3285708c01a5b6d480cp-1L, 325 0x1.dd321f301b4604b695de3c0630c0p-1L, 326 0x1.dfc97337b9b5eb968cac39ed284cp-1L, 327 0x1.e264614f5a128a12761fa17adc74p-1L, 328 0x1.e502ee78b3ff6273d130153992d0p-1L, 329 0x1.e7a51fbc74c834b548b2832378a4p-1L, 330 0x1.ea4afa2a490d9858f73a18f5dab4p-1L, 331 0x1.ecf482d8e67f08db0312fb949d50p-1L, 332 0x1.efa1bee615a27771fd21a92dabb6p-1L, 333 0x1.f252b376bba974e8696fc3638f24p-1L, 334 0x1.f50765b6e4540674f84b762861a6p-1L, 335 0x1.f7bfdad9cbe138913b4bfe72bd78p-1L, 336 0x1.fa7c1819e90d82e90a7e74b26360p-1L, 337 0x1.fd3c22b8f71f10975ba4b32bd006p-1L, 338 0x1.0000000000000000000000000000p+0L, 339 0x1.0163da9fb33356d84a66ae336e98p+0L, 340 0x1.02c9a3e778060ee6f7caca4f7a18p+0L, 341 0x1.04315e86e7f84bd738f9a20da442p+0L, 342 0x1.059b0d31585743ae7c548eb68c6ap+0L, 343 0x1.0706b29ddf6ddc6dc403a9d87b1ep+0L, 344 0x1.0874518759bc808c35f25d942856p+0L, 345 0x1.09e3ecac6f3834521e060c584d5cp+0L, 346 0x1.0b5586cf9890f6298b92b7184200p+0L, 347 0x1.0cc922b7247f7407b705b893dbdep+0L, 348 0x1.0e3ec32d3d1a2020742e4f8af794p+0L, 349 0x1.0fb66affed31af232091dd8a169ep+0L, 350 0x1.11301d0125b50a4ebbf1aed9321cp+0L, 351 0x1.12abdc06c31cbfb92bad324d6f84p+0L, 352 0x1.1429aaea92ddfb34101943b2588ep+0L, 353 0x1.15a98c8a58e512480d573dd562aep+0L, 354 0x1.172b83c7d517adcdf7c8c50eb162p+0L, 355 0x1.18af9388c8de9bbbf70b9a3c269cp+0L, 356 0x1.1a35beb6fcb753cb698f692d2038p+0L, 357 0x1.1bbe084045cd39ab1e72b442810ep+0L, 358 0x1.1d4873168b9aa7805b8028990be8p+0L, 359 0x1.1ed5022fcd91cb8819ff61121fbep+0L, 360 0x1.2063b88628cd63b8eeb0295093f6p+0L, 361 0x1.21f49917ddc962552fd29294bc20p+0L, 362 0x1.2387a6e75623866c1fadb1c159c0p+0L, 363 0x1.251ce4fb2a63f3582ab7de9e9562p+0L, 364 0x1.26b4565e27cdd257a673281d3068p+0L, 365 0x1.284dfe1f5638096cf15cf03c9fa0p+0L, 366 0x1.29e9df51fdee12c25d15f5a25022p+0L, 367 0x1.2b87fd0dad98ffddea46538fca24p+0L, 368 0x1.2d285a6e4030b40091d536d0733ep+0L, 369 0x1.2ecafa93e2f5611ca0f45d5239a4p+0L, 370 0x1.306fe0a31b7152de8d5a463063bep+0L, 371 0x1.32170fc4cd8313539cf1c3009330p+0L, 372 0x1.33c08b26416ff4c9c8610d96680ep+0L, 373 0x1.356c55f929ff0c94623476373be4p+0L, 374 0x1.371a7373aa9caa7145502f45452ap+0L, 375 0x1.38cae6d05d86585a9cb0d9bed530p+0L, 376 0x1.3a7db34e59ff6ea1bc9299e0a1fep+0L, 377 0x1.3c32dc313a8e484001f228b58cf0p+0L, 378 0x1.3dea64c12342235b41223e13d7eep+0L, 379 0x1.3fa4504ac801ba0bf701aa417b9cp+0L, 380 0x1.4160a21f72e29f84325b8f3dbacap+0L, 381 0x1.431f5d950a896dc704439410b628p+0L, 382 0x1.44e086061892d03136f409df0724p+0L, 383 0x1.46a41ed1d005772512f459229f0ap+0L, 384 0x1.486a2b5c13cd013c1a3b69062f26p+0L, 385 0x1.4a32af0d7d3de672d8bcf46f99b4p+0L, 386 0x1.4bfdad5362a271d4397afec42e36p+0L, 387 0x1.4dcb299fddd0d63b36ef1a9e19dep+0L, 388 0x1.4f9b2769d2ca6ad33d8b69aa0b8cp+0L, 389 0x1.516daa2cf6641c112f52c84d6066p+0L, 390 0x1.5342b569d4f81df0a83c49d86bf4p+0L, 391 0x1.551a4ca5d920ec52ec620243540cp+0L, 392 0x1.56f4736b527da66ecb004764e61ep+0L, 393 0x1.58d12d497c7fd252bc2b7343d554p+0L, 394 0x1.5ab07dd48542958c93015191e9a8p+0L, 395 0x1.5c9268a5946b701c4b1b81697ed4p+0L, 396 0x1.5e76f15ad21486e9be4c20399d12p+0L, 397 0x1.605e1b976dc08b076f592a487066p+0L, 398 0x1.6247eb03a5584b1f0fa06fd2d9eap+0L, 399 0x1.6434634ccc31fc76f8714c4ee122p+0L, 400 0x1.66238825522249127d9e29b92ea2p+0L, 401 0x1.68155d44ca973081c57227b9f69ep+0L, 402}; 403 404static const float eps[TBLSIZE] = { 405 -0x1.5c50p-101, 406 -0x1.5d00p-106, 407 0x1.8e90p-102, 408 -0x1.5340p-103, 409 0x1.1bd0p-102, 410 -0x1.4600p-105, 411 -0x1.7a40p-104, 412 0x1.d590p-102, 413 -0x1.d590p-101, 414 0x1.b100p-103, 415 -0x1.0d80p-105, 416 0x1.6b00p-103, 417 -0x1.9f00p-105, 418 0x1.c400p-103, 419 0x1.e120p-103, 420 -0x1.c100p-104, 421 -0x1.9d20p-103, 422 0x1.a800p-108, 423 0x1.4c00p-106, 424 -0x1.9500p-106, 425 0x1.6900p-105, 426 -0x1.29d0p-100, 427 0x1.4c60p-103, 428 0x1.13a0p-102, 429 -0x1.5b60p-103, 430 -0x1.1c40p-103, 431 0x1.db80p-102, 432 0x1.91a0p-102, 433 0x1.dc00p-105, 434 0x1.44c0p-104, 435 0x1.9710p-102, 436 0x1.8760p-103, 437 -0x1.a720p-103, 438 0x1.ed20p-103, 439 -0x1.49c0p-102, 440 -0x1.e000p-111, 441 0x1.86a0p-103, 442 0x1.2b40p-103, 443 -0x1.b400p-108, 444 0x1.1280p-99, 445 -0x1.02d8p-102, 446 -0x1.e3d0p-103, 447 -0x1.b080p-105, 448 -0x1.f100p-107, 449 -0x1.16c0p-105, 450 -0x1.1190p-103, 451 -0x1.a7d2p-100, 452 0x1.3450p-103, 453 -0x1.67c0p-105, 454 0x1.4b80p-104, 455 -0x1.c4e0p-103, 456 0x1.6000p-108, 457 -0x1.3f60p-105, 458 0x1.93f0p-104, 459 0x1.5fe0p-105, 460 0x1.6f80p-107, 461 -0x1.7600p-106, 462 0x1.21e0p-106, 463 -0x1.3a40p-106, 464 -0x1.40c0p-104, 465 -0x1.9860p-105, 466 -0x1.5d40p-108, 467 -0x1.1d70p-106, 468 0x1.2760p-105, 469 0x0.0000p+0, 470 0x1.21e2p-104, 471 -0x1.9520p-108, 472 -0x1.5720p-106, 473 -0x1.4810p-106, 474 -0x1.be00p-109, 475 0x1.0080p-105, 476 -0x1.5780p-108, 477 -0x1.d460p-105, 478 -0x1.6140p-105, 479 0x1.4630p-104, 480 0x1.ad50p-103, 481 0x1.82e0p-105, 482 0x1.1d3cp-101, 483 0x1.6100p-107, 484 0x1.ec30p-104, 485 0x1.f200p-108, 486 0x1.0b40p-103, 487 0x1.3660p-102, 488 0x1.d9d0p-103, 489 -0x1.02d0p-102, 490 0x1.b070p-103, 491 0x1.b9c0p-104, 492 -0x1.01c0p-103, 493 -0x1.dfe0p-103, 494 0x1.1b60p-104, 495 -0x1.ae94p-101, 496 -0x1.3340p-104, 497 0x1.b3d8p-102, 498 -0x1.6e40p-105, 499 -0x1.3670p-103, 500 0x1.c140p-104, 501 0x1.1840p-101, 502 0x1.1ab0p-102, 503 -0x1.a400p-104, 504 0x1.1f00p-104, 505 -0x1.7180p-103, 506 0x1.4ce0p-102, 507 0x1.9200p-107, 508 -0x1.54c0p-103, 509 0x1.1b80p-105, 510 -0x1.1828p-101, 511 0x1.5720p-102, 512 -0x1.a060p-100, 513 0x1.9160p-102, 514 0x1.a280p-104, 515 0x1.3400p-107, 516 0x1.2b20p-102, 517 0x1.7800p-108, 518 0x1.cfd0p-101, 519 0x1.2ef0p-102, 520 -0x1.2760p-99, 521 0x1.b380p-104, 522 0x1.0048p-101, 523 -0x1.60b0p-102, 524 0x1.a1ccp-100, 525 -0x1.a640p-104, 526 -0x1.08a0p-101, 527 0x1.7e60p-102, 528 0x1.22c0p-103, 529 -0x1.7200p-106, 530 0x1.f0f0p-102, 531 0x1.eb4ep-99, 532 0x1.c6e0p-103, 533}; 534 535/* 536 * exp2l(x): compute the base 2 exponential of x 537 * 538 * Accuracy: Peak error < 0.502 ulp. 539 * 540 * Method: (accurate tables) 541 * 542 * Reduce x: 543 * x = 2**k + y, for integer k and |y| <= 1/2. 544 * Thus we have exp2(x) = 2**k * exp2(y). 545 * 546 * Reduce y: 547 * y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE. 548 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]), 549 * with |z - eps[i]| <= 2**-8 + 2**-98 for the table used. 550 * 551 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via 552 * a degree-10 minimax polynomial with maximum error under 2**-120. 553 * The values in exp2t[] and eps[] are chosen such that 554 * exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such 555 * that exp2t[i] is accurate to 2**-122. 556 * 557 * Note that the range of i is +-TBLSIZE/2, so we actually index the tables 558 * by i0 = i + TBLSIZE/2. 559 * 560 * This method is due to Gal, with many details due to Gal and Bachelis: 561 * 562 * Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library 563 * for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991). 564 */ 565long double 566exp2l(long double x) 567{ 568 union ldshape u = {x}; 569 int e = u.i.se & 0x7fff; 570 long double r, z, t; 571 uint32_t i0; 572 union {uint32_t u; int32_t i;} k; 573 574 /* Filter out exceptional cases. */ 575 if (e >= 0x3fff + 14) { /* |x| >= 16384 or x is NaN */ 576 if (u.i.se >= 0x3fff + 15 && u.i.se >> 15 == 0) 577 /* overflow */ 578 return x * 0x1p16383L; 579 if (e == 0x7fff) /* -inf or -nan */ 580 return -1/x; 581 if (x < -16382) { 582 if (x <= -16495 || x - 0x1p112 + 0x1p112 != x) 583 /* underflow */ 584 FORCE_EVAL((float)(-0x1p-149/x)); 585 if (x <= -16446) 586 return 0; 587 } 588 } else if (e < 0x3fff - 114) { 589 return 1 + x; 590 } 591 592 /* 593 * Reduce x, computing z, i0, and k. The low bits of x + redux 594 * contain the 16-bit integer part of the exponent (k) followed by 595 * TBLBITS fractional bits (i0). We use bit tricks to extract these 596 * as integers, then set z to the remainder. 597 * 598 * Example: Suppose x is 0xabc.123456p0 and TBLBITS is 8. 599 * Then the low-order word of x + redux is 0x000abc12, 600 * We split this into k = 0xabc and i0 = 0x12 (adjusted to 601 * index into the table), then we compute z = 0x0.003456p0. 602 */ 603 u.f = x + redux; 604 i0 = u.i2.lo + TBLSIZE / 2; 605 k.u = i0 / TBLSIZE * TBLSIZE; 606 k.i /= TBLSIZE; 607 i0 %= TBLSIZE; 608 u.f -= redux; 609 z = x - u.f; 610 611 /* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */ 612 t = tbl[i0]; 613 z -= eps[i0]; 614 r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 + z * (P6 615 + z * (P7 + z * (P8 + z * (P9 + z * P10))))))))); 616 617 return scalbnl(r, k.i); 618} 619#endif 620