1 /*- 2 * Copyright (c) 2009-2013 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 * ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments. 34 */ 35 36 #include <float.h> 37 38 #include "fpmath.h" 39 #include "math.h" 40 #include "math_private.h" 41 42 #define INTERVALS 128 43 #define LOG2_INTERVALS 7 44 #define BIAS (LDBL_MAX_EXP - 1) 45 46 static const long double 47 huge = 0x1p10000L, 48 twom10000 = 0x1p-10000L; 49 /* XXX Prevent gcc from erroneously constant folding this: */ 50 static volatile const long double tiny = 0x1p-10000L; 51 52 static const long double 53 /* log(2**16384 - 0.5) rounded towards zero: */ 54 /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ 55 o_threshold = 11356.523406294143949491931077970763428L, 56 /* log(2**(-16381-64-1)) rounded towards zero: */ 57 u_threshold = -11433.462743336297878837243843452621503L; 58 59 static const double 60 /* 61 * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must 62 * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest 63 * bits zero so that multiplication of it by n is exact. 64 */ 65 INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ 66 L2 = -1.0253670638894731e-29; /* -0x1.9ff0342542fc3p-97 */ 67 static const long double 68 /* 0x1.62e42fefa39ef35793c768000000p-8 */ 69 L1 = 5.41521234812457272982212595914567508e-3L; 70 71 static const long double 72 /* 73 * Domain [-0.002708, 0.002708], range ~[-2.4021e-38, 2.4234e-38]: 74 * |exp(x) - p(x)| < 2**-124.9 75 * (0.002708 is ln2/(2*INTERVALS) rounded up a little). 76 */ 77 A2 = 0.5, 78 A3 = 1.66666666666666666666666666651085500e-1L, 79 A4 = 4.16666666666666666666666666425885320e-2L, 80 A5 = 8.33333333333333333334522877160175842e-3L, 81 A6 = 1.38888888888888888889971139751596836e-3L; 82 83 static const double 84 A7 = 1.9841269841269471e-4, 85 A8 = 2.4801587301585284e-5, 86 A9 = 2.7557324277411234e-6, 87 A10 = 2.7557333722375072e-7; 88 89 static const struct { 90 /* 91 * hi must be rounded to at most 106 bits so that multiplication 92 * by r1 in expm1l() is exact, but it is rounded to 88 bits due to 93 * historical accidents. 94 */ 95 long double hi; 96 long double lo; 97 } tbl[INTERVALS] = { 98 0x1p0L, 0x0p0L, 99 0x1.0163da9fb33356d84a66aep0L, 0x3.36dcdfa4003ec04c360be2404078p-92L, 100 0x1.02c9a3e778060ee6f7cacap0L, 0x4.f7a29bde93d70a2cabc5cb89ba10p-92L, 101 0x1.04315e86e7f84bd738f9a2p0L, 0xd.a47e6ed040bb4bfc05af6455e9b8p-96L, 102 0x1.059b0d31585743ae7c548ep0L, 0xb.68ca417fe53e3495f7df4baf84a0p-92L, 103 0x1.0706b29ddf6ddc6dc403a8p0L, 0x1.d87b27ed07cb8b092ac75e311753p-88L, 104 0x1.0874518759bc808c35f25cp0L, 0x1.9427fa2b041b2d6829d8993a0d01p-88L, 105 0x1.09e3ecac6f3834521e060cp0L, 0x5.84d6b74ba2e023da730e7fccb758p-92L, 106 0x1.0b5586cf9890f6298b92b6p0L, 0x1.1842a98364291408b3ceb0a2a2bbp-88L, 107 0x1.0cc922b7247f7407b705b8p0L, 0x9.3dc5e8aac564e6fe2ef1d431fd98p-92L, 108 0x1.0e3ec32d3d1a2020742e4ep0L, 0x1.8af6a552ac4b358b1129e9f966a4p-88L, 109 0x1.0fb66affed31af232091dcp0L, 0x1.8a1426514e0b627bda694a400a27p-88L, 110 0x1.11301d0125b50a4ebbf1aep0L, 0xd.9318ceac5cc47ab166ee57427178p-92L, 111 0x1.12abdc06c31cbfb92bad32p0L, 0x4.d68e2f7270bdf7cedf94eb1cb818p-92L, 112 0x1.1429aaea92ddfb34101942p0L, 0x1.b2586d01844b389bea7aedd221d4p-88L, 113 0x1.15a98c8a58e512480d573cp0L, 0x1.d5613bf92a2b618ee31b376c2689p-88L, 114 0x1.172b83c7d517adcdf7c8c4p0L, 0x1.0eb14a792035509ff7d758693f24p-88L, 115 0x1.18af9388c8de9bbbf70b9ap0L, 0x3.c2505c97c0102e5f1211941d2840p-92L, 116 0x1.1a35beb6fcb753cb698f68p0L, 0x1.2d1c835a6c30724d5cfae31b84e5p-88L, 117 0x1.1bbe084045cd39ab1e72b4p0L, 0x4.27e35f9acb57e473915519a1b448p-92L, 118 0x1.1d4873168b9aa7805b8028p0L, 0x9.90f07a98b42206e46166cf051d70p-92L, 119 0x1.1ed5022fcd91cb8819ff60p0L, 0x1.121d1e504d36c47474c9b7de6067p-88L, 120 0x1.2063b88628cd63b8eeb028p0L, 0x1.50929d0fc487d21c2b84004264dep-88L, 121 0x1.21f49917ddc962552fd292p0L, 0x9.4bdb4b61ea62477caa1dce823ba0p-92L, 122 0x1.2387a6e75623866c1fadb0p0L, 0x1.c15cb593b0328566902df69e4de2p-88L, 123 0x1.251ce4fb2a63f3582ab7dep0L, 0x9.e94811a9c8afdcf796934bc652d0p-92L, 124 0x1.26b4565e27cdd257a67328p0L, 0x1.d3b249dce4e9186ddd5ff44e6b08p-92L, 125 0x1.284dfe1f5638096cf15cf0p0L, 0x3.ca0967fdaa2e52d7c8106f2e262cp-92L, 126 0x1.29e9df51fdee12c25d15f4p0L, 0x1.a24aa3bca890ac08d203fed80a07p-88L, 127 0x1.2b87fd0dad98ffddea4652p0L, 0x1.8fcab88442fdc3cb6de4519165edp-88L, 128 0x1.2d285a6e4030b40091d536p0L, 0xd.075384589c1cd1b3e4018a6b1348p-92L, 129 0x1.2ecafa93e2f5611ca0f45cp0L, 0x1.523833af611bdcda253c554cf278p-88L, 130 0x1.306fe0a31b7152de8d5a46p0L, 0x3.05c85edecbc27343629f502f1af2p-92L, 131 0x1.32170fc4cd8313539cf1c2p0L, 0x1.008f86dde3220ae17a005b6412bep-88L, 132 0x1.33c08b26416ff4c9c8610cp0L, 0x1.96696bf95d1593039539d94d662bp-88L, 133 0x1.356c55f929ff0c94623476p0L, 0x3.73af38d6d8d6f9506c9bbc93cbc0p-92L, 134 0x1.371a7373aa9caa7145502ep0L, 0x1.4547987e3e12516bf9c699be432fp-88L, 135 0x1.38cae6d05d86585a9cb0d8p0L, 0x1.bed0c853bd30a02790931eb2e8f0p-88L, 136 0x1.3a7db34e59ff6ea1bc9298p0L, 0x1.e0a1d336163fe2f852ceeb134067p-88L, 137 0x1.3c32dc313a8e484001f228p0L, 0xb.58f3775e06ab66353001fae9fca0p-92L, 138 0x1.3dea64c12342235b41223ep0L, 0x1.3d773fba2cb82b8244267c54443fp-92L, 139 0x1.3fa4504ac801ba0bf701aap0L, 0x4.1832fb8c1c8dbdff2c49909e6c60p-92L, 140 0x1.4160a21f72e29f84325b8ep0L, 0x1.3db61fb352f0540e6ba05634413ep-88L, 141 0x1.431f5d950a896dc7044394p0L, 0x1.0ccec81e24b0caff7581ef4127f7p-92L, 142 0x1.44e086061892d03136f408p0L, 0x1.df019fbd4f3b48709b78591d5cb5p-88L, 143 0x1.46a41ed1d005772512f458p0L, 0x1.229d97df404ff21f39c1b594d3a8p-88L, 144 0x1.486a2b5c13cd013c1a3b68p0L, 0x1.062f03c3dd75ce8757f780e6ec99p-88L, 145 0x1.4a32af0d7d3de672d8bcf4p0L, 0x6.f9586461db1d878b1d148bd3ccb8p-92L, 146 0x1.4bfdad5362a271d4397afep0L, 0xc.42e20e0363ba2e159c579f82e4b0p-92L, 147 0x1.4dcb299fddd0d63b36ef1ap0L, 0x9.e0cc484b25a5566d0bd5f58ad238p-92L, 148 0x1.4f9b2769d2ca6ad33d8b68p0L, 0x1.aa073ee55e028497a329a7333dbap-88L, 149 0x1.516daa2cf6641c112f52c8p0L, 0x4.d822190e718226177d7608d20038p-92L, 150 0x1.5342b569d4f81df0a83c48p0L, 0x1.d86a63f4e672a3e429805b049465p-88L, 151 0x1.551a4ca5d920ec52ec6202p0L, 0x4.34ca672645dc6c124d6619a87574p-92L, 152 0x1.56f4736b527da66ecb0046p0L, 0x1.64eb3c00f2f5ab3d801d7cc7272dp-88L, 153 0x1.58d12d497c7fd252bc2b72p0L, 0x1.43bcf2ec936a970d9cc266f0072fp-88L, 154 0x1.5ab07dd48542958c930150p0L, 0x1.91eb345d88d7c81280e069fbdb63p-88L, 155 0x1.5c9268a5946b701c4b1b80p0L, 0x1.6986a203d84e6a4a92f179e71889p-88L, 156 0x1.5e76f15ad21486e9be4c20p0L, 0x3.99766a06548a05829e853bdb2b52p-92L, 157 0x1.605e1b976dc08b076f592ap0L, 0x4.86e3b34ead1b4769df867b9c89ccp-92L, 158 0x1.6247eb03a5584b1f0fa06ep0L, 0x1.d2da42bb1ceaf9f732275b8aef30p-88L, 159 0x1.6434634ccc31fc76f8714cp0L, 0x4.ed9a4e41000307103a18cf7a6e08p-92L, 160 0x1.66238825522249127d9e28p0L, 0x1.b8f314a337f4dc0a3adf1787ff74p-88L, 161 0x1.68155d44ca973081c57226p0L, 0x1.b9f32706bfe4e627d809a85dcc66p-88L, 162 0x1.6a09e667f3bcc908b2fb12p0L, 0x1.66ea957d3e3adec17512775099dap-88L, 163 0x1.6c012750bdabeed76a9980p0L, 0xf.4f33fdeb8b0ecd831106f57b3d00p-96L, 164 0x1.6dfb23c651a2ef220e2cbep0L, 0x1.bbaa834b3f11577ceefbe6c1c411p-92L, 165 0x1.6ff7df9519483cf87e1b4ep0L, 0x1.3e213bff9b702d5aa477c12523cep-88L, 166 0x1.71f75e8ec5f73dd2370f2ep0L, 0xf.0acd6cb434b562d9e8a20adda648p-92L, 167 0x1.73f9a48a58173bd5c9a4e6p0L, 0x8.ab1182ae217f3a7681759553e840p-92L, 168 0x1.75feb564267c8bf6e9aa32p0L, 0x1.a48b27071805e61a17b954a2dad8p-88L, 169 0x1.780694fde5d3f619ae0280p0L, 0x8.58b2bb2bdcf86cd08e35fb04c0f0p-92L, 170 0x1.7a11473eb0186d7d51023ep0L, 0x1.6cda1f5ef42b66977960531e821bp-88L, 171 0x1.7c1ed0130c1327c4933444p0L, 0x1.937562b2dc933d44fc828efd4c9cp-88L, 172 0x1.7e2f336cf4e62105d02ba0p0L, 0x1.5797e170a1427f8fcdf5f3906108p-88L, 173 0x1.80427543e1a11b60de6764p0L, 0x9.a354ea706b8e4d8b718a672bf7c8p-92L, 174 0x1.82589994cce128acf88afap0L, 0xb.34a010f6ad65cbbac0f532d39be0p-92L, 175 0x1.8471a4623c7acce52f6b96p0L, 0x1.c64095370f51f48817914dd78665p-88L, 176 0x1.868d99b4492ec80e41d90ap0L, 0xc.251707484d73f136fb5779656b70p-92L, 177 0x1.88ac7d98a669966530bcdep0L, 0x1.2d4e9d61283ef385de170ab20f96p-88L, 178 0x1.8ace5422aa0db5ba7c55a0p0L, 0x1.92c9bb3e6ed61f2733304a346d8fp-88L, 179 0x1.8cf3216b5448bef2aa1cd0p0L, 0x1.61c55d84a9848f8c453b3ca8c946p-88L, 180 0x1.8f1ae991577362b982745cp0L, 0x7.2ed804efc9b4ae1458ae946099d4p-92L, 181 0x1.9145b0b91ffc588a61b468p0L, 0x1.f6b70e01c2a90229a4c4309ea719p-88L, 182 0x1.93737b0cdc5e4f4501c3f2p0L, 0x5.40a22d2fc4af581b63e8326efe9cp-92L, 183 0x1.95a44cbc8520ee9b483694p0L, 0x1.a0fc6f7c7d61b2b3a22a0eab2cadp-88L, 184 0x1.97d829fde4e4f8b9e920f8p0L, 0x1.1e8bd7edb9d7144b6f6818084cc7p-88L, 185 0x1.9a0f170ca07b9ba3109b8cp0L, 0x4.6737beb19e1eada6825d3c557428p-92L, 186 0x1.9c49182a3f0901c7c46b06p0L, 0x1.1f2be58ddade50c217186c90b457p-88L, 187 0x1.9e86319e323231824ca78ep0L, 0x6.4c6e010f92c082bbadfaf605cfd4p-92L, 188 0x1.a0c667b5de564b29ada8b8p0L, 0xc.ab349aa0422a8da7d4512edac548p-92L, 189 0x1.a309bec4a2d3358c171f76p0L, 0x1.0daad547fa22c26d168ea762d854p-88L, 190 0x1.a5503b23e255c8b424491cp0L, 0xa.f87bc8050a405381703ef7caff50p-92L, 191 0x1.a799e1330b3586f2dfb2b0p0L, 0x1.58f1a98796ce8908ae852236ca94p-88L, 192 0x1.a9e6b5579fdbf43eb243bcp0L, 0x1.ff4c4c58b571cf465caf07b4b9f5p-88L, 193 0x1.ac36bbfd3f379c0db966a2p0L, 0x1.1265fc73e480712d20f8597a8e7bp-88L, 194 0x1.ae89f995ad3ad5e8734d16p0L, 0x1.73205a7fbc3ae675ea440b162d6cp-88L, 195 0x1.b0e07298db66590842acdep0L, 0x1.c6f6ca0e5dcae2aafffa7a0554cbp-88L, 196 0x1.b33a2b84f15faf6bfd0e7ap0L, 0x1.d947c2575781dbb49b1237c87b6ep-88L, 197 0x1.b59728de559398e3881110p0L, 0x1.64873c7171fefc410416be0a6525p-88L, 198 0x1.b7f76f2fb5e46eaa7b081ap0L, 0xb.53c5354c8903c356e4b625aacc28p-92L, 199 0x1.ba5b030a10649840cb3c6ap0L, 0xf.5b47f297203757e1cc6eadc8bad0p-92L, 200 0x1.bcc1e904bc1d2247ba0f44p0L, 0x1.b3d08cd0b20287092bd59be4ad98p-88L, 201 0x1.bf2c25bd71e088408d7024p0L, 0x1.18e3449fa073b356766dfb568ff4p-88L, 202 0x1.c199bdd85529c2220cb12ap0L, 0x9.1ba6679444964a36661240043970p-96L, 203 0x1.c40ab5fffd07a6d14df820p0L, 0xf.1828a5366fd387a7bdd54cdf7300p-92L, 204 0x1.c67f12e57d14b4a2137fd2p0L, 0xf.2b301dd9e6b151a6d1f9d5d5f520p-96L, 205 0x1.c8f6d9406e7b511acbc488p0L, 0x5.c442ddb55820171f319d9e5076a8p-96L, 206 0x1.cb720dcef90691503cbd1ep0L, 0x9.49db761d9559ac0cb6dd3ed599e0p-92L, 207 0x1.cdf0b555dc3f9c44f8958ep0L, 0x1.ac51be515f8c58bdfb6f5740a3a4p-88L, 208 0x1.d072d4a07897b8d0f22f20p0L, 0x1.a158e18fbbfc625f09f4cca40874p-88L, 209 0x1.d2f87080d89f18ade12398p0L, 0x9.ea2025b4c56553f5cdee4c924728p-92L, 210 0x1.d5818dcfba48725da05aeap0L, 0x1.66e0dca9f589f559c0876ff23830p-88L, 211 0x1.d80e316c98397bb84f9d04p0L, 0x8.805f84bec614de269900ddf98d28p-92L, 212 0x1.da9e603db3285708c01a5ap0L, 0x1.6d4c97f6246f0ec614ec95c99392p-88L, 213 0x1.dd321f301b4604b695de3cp0L, 0x6.30a393215299e30d4fb73503c348p-96L, 214 0x1.dfc97337b9b5eb968cac38p0L, 0x1.ed291b7225a944efd5bb5524b927p-88L, 215 0x1.e264614f5a128a12761fa0p0L, 0x1.7ada6467e77f73bf65e04c95e29dp-88L, 216 0x1.e502ee78b3ff6273d13014p0L, 0x1.3991e8f49659e1693be17ae1d2f9p-88L, 217 0x1.e7a51fbc74c834b548b282p0L, 0x1.23786758a84f4956354634a416cep-88L, 218 0x1.ea4afa2a490d9858f73a18p0L, 0xf.5db301f86dea20610ceee13eb7b8p-92L, 219 0x1.ecf482d8e67f08db0312fap0L, 0x1.949cef462010bb4bc4ce72a900dfp-88L, 220 0x1.efa1bee615a27771fd21a8p0L, 0x1.2dac1f6dd5d229ff68e46f27e3dfp-88L, 221 0x1.f252b376bba974e8696fc2p0L, 0x1.6390d4c6ad5476b5162f40e1d9a9p-88L, 222 0x1.f50765b6e4540674f84b76p0L, 0x2.862baff99000dfc4352ba29b8908p-92L, 223 0x1.f7bfdad9cbe138913b4bfep0L, 0x7.2bd95c5ce7280fa4d2344a3f5618p-92L, 224 0x1.fa7c1819e90d82e90a7e74p0L, 0xb.263c1dc060c36f7650b4c0f233a8p-92L, 225 0x1.fd3c22b8f71f10975ba4b2p0L, 0x1.2bcf3a5e12d269d8ad7c1a4a8875p-88L 226 }; 227 228 long double 229 expl(long double x) 230 { 231 union IEEEl2bits u, v; 232 long double q, r, r1, t, twopk, twopkp10000; 233 double dr, fn, r2; 234 int k, n, n2; 235 uint16_t hx, ix; 236 237 /* Filter out exceptional cases. */ 238 u.e = x; 239 hx = u.xbits.expsign; 240 ix = hx & 0x7fff; 241 if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ 242 if (ix == BIAS + LDBL_MAX_EXP) { 243 if (hx & 0x8000) /* x is -Inf or -NaN */ 244 return (-1 / x); 245 return (x + x); /* x is +Inf or +NaN */ 246 } 247 if (x > o_threshold) 248 return (huge * huge); 249 if (x < u_threshold) 250 return (tiny * tiny); 251 } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */ 252 return (1 + x); /* 1 with inexact iff x != 0 */ 253 } 254 255 ENTERI(); 256 257 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 258 /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 259 /* XXX assume no extra precision for the additions, as for trig fns. */ 260 /* XXX this set of comments is now quadruplicated. */ 261 fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52; 262 #if defined(HAVE_EFFICIENT_IRINT) 263 n = irint(fn); 264 #else 265 n = (int)fn; 266 #endif 267 n2 = (unsigned)n % INTERVALS; 268 k = n >> LOG2_INTERVALS; 269 r1 = x - fn * L1; 270 r2 = fn * -L2; 271 r = r1 + r2; 272 273 /* Prepare scale factors. */ 274 /* XXX sparc64 multiplication is so slow that scalbnl() is faster. */ 275 v.e = 1; 276 if (k >= LDBL_MIN_EXP) { 277 v.xbits.expsign = BIAS + k; 278 twopk = v.e; 279 } else { 280 v.xbits.expsign = BIAS + k + 10000; 281 twopkp10000 = v.e; 282 } 283 284 /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */ 285 dr = r; 286 q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + 287 dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); 288 t = tbl[n2].lo + tbl[n2].hi; 289 t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; 290 291 /* Scale by 2**k. */ 292 if (k >= LDBL_MIN_EXP) { 293 if (k == LDBL_MAX_EXP) 294 RETURNI(t * 2 * 0x1p16383L); 295 RETURNI(t * twopk); 296 } else { 297 RETURNI(t * twopkp10000 * twom10000); 298 } 299 } 300 301 /* 302 * Our T1 and T2 are chosen to be approximately the points where method 303 * A and method B have the same accuracy. Tang's T1 and T2 are the 304 * points where method A's accuracy changes by a full bit. For Tang, 305 * this drop in accuracy makes method A immediately less accurate than 306 * method B, but our larger INTERVALS makes method A 2 bits more 307 * accurate so it remains the most accurate method significantly 308 * closer to the origin despite losing the full bit in our extended 309 * range for it. 310 * 311 * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2]. 312 * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear 313 * in both subintervals, so set T3 = 2**-5, which places the condition 314 * into the [T1, T3] interval. 315 */ 316 static const double 317 T1 = -0.1659, /* ~-30.625/128 * log(2) */ 318 T2 = 0.1659, /* ~30.625/128 * log(2) */ 319 T3 = 0.03125; 320 321 /* 322 * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]: 323 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03 324 */ 325 static const long double 326 C3 = 1.66666666666666666666666666666666667e-1L, 327 C4 = 4.16666666666666666666666666666666645e-2L, 328 C5 = 8.33333333333333333333333333333371638e-3L, 329 C6 = 1.38888888888888888888888888891188658e-3L, 330 C7 = 1.98412698412698412698412697235950394e-4L, 331 C8 = 2.48015873015873015873015112487849040e-5L, 332 C9 = 2.75573192239858906525606685484412005e-6L, 333 C10 = 2.75573192239858906612966093057020362e-7L, 334 C11 = 2.50521083854417203619031960151253944e-8L, 335 C12 = 2.08767569878679576457272282566520649e-9L, 336 C13 = 1.60590438367252471783548748824255707e-10L; 337 338 static const double 339 C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */ 340 C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */ 341 C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */ 342 C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */ 343 C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */ 344 345 /* 346 * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]: 347 * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44 348 */ 349 static const long double 350 D3 = 1.66666666666666666666666666666682245e-1L, 351 D4 = 4.16666666666666666666666666634228324e-2L, 352 D5 = 8.33333333333333333333333364022244481e-3L, 353 D6 = 1.38888888888888888888887138722762072e-3L, 354 D7 = 1.98412698412698412699085805424661471e-4L, 355 D8 = 2.48015873015873015687993712101479612e-5L, 356 D9 = 2.75573192239858944101036288338208042e-6L, 357 D10 = 2.75573192239853161148064676533754048e-7L, 358 D11 = 2.50521083855084570046480450935267433e-8L, 359 D12 = 2.08767569819738524488686318024854942e-9L, 360 D13 = 1.60590442297008495301927448122499313e-10L; 361 362 static const double 363 D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */ 364 D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */ 365 D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */ 366 D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */ 367 368 long double 369 expm1l(long double x) 370 { 371 union IEEEl2bits u, v; 372 long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi; 373 long double x_lo, x2; 374 double dr, dx, fn, r2; 375 int k, n, n2; 376 uint16_t hx, ix; 377 378 /* Filter out exceptional cases. */ 379 u.e = x; 380 hx = u.xbits.expsign; 381 ix = hx & 0x7fff; 382 if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */ 383 if (ix == BIAS + LDBL_MAX_EXP) { 384 if (hx & 0x8000) /* x is -Inf or -NaN */ 385 return (-1 / x - 1); 386 return (x + x); /* x is +Inf or +NaN */ 387 } 388 if (x > o_threshold) 389 return (huge * huge); 390 /* 391 * expm1l() never underflows, but it must avoid 392 * unrepresentable large negative exponents. We used a 393 * much smaller threshold for large |x| above than in 394 * expl() so as to handle not so large negative exponents 395 * in the same way as large ones here. 396 */ 397 if (hx & 0x8000) /* x <= -128 */ 398 return (tiny - 1); /* good for x < -114ln2 - eps */ 399 } 400 401 ENTERI(); 402 403 if (T1 < x && x < T2) { 404 x2 = x * x; 405 dx = x; 406 407 if (x < T3) { 408 if (ix < BIAS - 113) { /* |x| < 0x1p-113 */ 409 /* x (rounded) with inexact if x != 0: */ 410 RETURNI(x == 0 ? x : 411 (0x1p200 * x + fabsl(x)) * 0x1p-200); 412 } 413 q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 + 414 x * (C7 + x * (C8 + x * (C9 + x * (C10 + 415 x * (C11 + x * (C12 + x * (C13 + 416 dx * (C14 + dx * (C15 + dx * (C16 + 417 dx * (C17 + dx * C18)))))))))))))); 418 } else { 419 q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 + 420 x * (D7 + x * (D8 + x * (D9 + x * (D10 + 421 x * (D11 + x * (D12 + x * (D13 + 422 dx * (D14 + dx * (D15 + dx * (D16 + 423 dx * D17))))))))))))); 424 } 425 426 x_hi = (float)x; 427 x_lo = x - x_hi; 428 hx2_hi = x_hi * x_hi / 2; 429 hx2_lo = x_lo * (x + x_hi) / 2; 430 if (ix >= BIAS - 7) 431 RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi)); 432 else 433 RETURNI(hx2_lo + q + hx2_hi + x); 434 } 435 436 /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ 437 /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 438 fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52; 439 #if defined(HAVE_EFFICIENT_IRINT) 440 n = irint(fn); 441 #else 442 n = (int)fn; 443 #endif 444 n2 = (unsigned)n % INTERVALS; 445 k = n >> LOG2_INTERVALS; 446 r1 = x - fn * L1; 447 r2 = fn * -L2; 448 r = r1 + r2; 449 450 /* Prepare scale factor. */ 451 v.e = 1; 452 v.xbits.expsign = BIAS + k; 453 twopk = v.e; 454 455 /* 456 * Evaluate lower terms of 457 * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). 458 */ 459 dr = r; 460 q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + 461 dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); 462 463 t = tbl[n2].lo + tbl[n2].hi; 464 465 if (k == 0) { 466 t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + 467 (tbl[n2].hi - 1); 468 RETURNI(t); 469 } 470 if (k == -1) { 471 t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + 472 (tbl[n2].hi - 2); 473 RETURNI(t / 2); 474 } 475 if (k < -7) { 476 t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; 477 RETURNI(t * twopk - 1); 478 } 479 if (k > 2 * LDBL_MANT_DIG - 1) { 480 t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; 481 if (k == LDBL_MAX_EXP) 482 RETURNI(t * 2 * 0x1p16383L - 1); 483 RETURNI(t * twopk - 1); 484 } 485 486 v.xbits.expsign = BIAS - k; 487 twomk = v.e; 488 489 if (k > LDBL_MANT_DIG - 1) 490 t = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi; 491 else 492 t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk); 493 RETURNI(t * twopk); 494 } 495
