lib/msun/ld128/s_expl.c

   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 }