src/math/exp2l.c

   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
  31 long 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
  39 static const double
  40 redux = 0x1.8p63 / TBLSIZE,
  41 P1    = 0x1.62e42fefa39efp-1,
  42 P2    = 0x1.ebfbdff82c58fp-3,
  43 P3    = 0x1.c6b08d7049fap-5,
  44 P4    = 0x1.3b2ab6fba4da5p-7,
  45 P5    = 0x1.5d8804780a736p-10,
  46 P6    = 0x1.430918835e33dp-13;
  47
  48 static 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  */
 200 long 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
 258 static 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
 266 static 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
 273 static 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
 404 static 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  */
 565 long double
 566 exp2l(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