lib/msun/ld80/s_expl.c

   1 /*-
   2  * Copyright (c) 2009-2012 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  * Compute the exponential of x for Intel 80-bit format.  This is based on:
  34  *
  35  *   PTP Tang, "Table-driven implementation of the exponential function
  36  *   in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15,
  37  *   144-157 (1989).
  38  *
  39  * where the 32 table entries have been expanded to INTERVALS (see below).
  40  */
  41
  42 #include <float.h>
  43
  44 #ifdef __i386__
  45 #include <ieeefp.h>
  46 #endif
  47
  48 #include "fpmath.h"
  49 #include "math.h"
  50 #include "math_private.h"
  51
  52 #define BIAS    (LDBL_MAX_EXP - 1)
  53
  54 static const long double
  55 huge = 0x1p10000L,
  56 twom10000 = 0x1p-10000L;
  57 /* XXX Prevent gcc from erroneously constant folding this: */
  58 static volatile const long double tiny = 0x1p-10000L;
  59
  60 static const union IEEEl2bits
  61 /* log(2**16384 - 0.5) rounded towards zero: */
  62 o_threshold = LD80C(0xb17217f7d1cf79ab, 13, 0,  11356.5234062941439488L),
  63 /* log(2**(-16381-64-1)) rounded towards zero: */
  64 u_threshold = LD80C(0xb21dfe7f09e2baa9, 13, 1, -11399.4985314888605581L);
  65
  66 static const double __aligned(64)
  67 /*
  68  * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication).  L1 must
  69  * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest
  70  * bits zero so that multiplication of it by n is exact.
  71  */
  72 L1 =  5.4152123484527692e-3,            /*  0x162e42ff000000.0p-60 */
  73 L2 = -3.2819649005320973e-13,           /* -0x1718432a1b0e26.0p-94 */
  74 INV_L = 1.8466496523378731e+2,          /*  0x171547652b82fe.0p-45 */
  75 /*
  76  * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]:
  77  * |exp(x) - p(x)| < 2**-77.2
  78  * (0.002708 is ln2/(2*INTERVALS) rounded up a little).
  79  */
  80 P2 =  0.5,
  81 P3 =  1.6666666666666119e-1,            /*  0x15555555555490.0p-55 */
  82 P4 =  4.1666666666665887e-2,            /*  0x155555555554e5.0p-57 */
  83 P5 =  8.3333354987869413e-3,            /*  0x1111115b789919.0p-59 */
  84 P6 =  1.3888891738560272e-3;            /*  0x16c16c651633ae.0p-62 */
  85
  86 /*
  87  * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where
  88  * the first 47 (?!) bits of the significand is stored in hi and the next 53
  89  * bits are in lo.
  90  */
  91 #define INTERVALS       128
  92
  93 static const struct {
  94         double  hi;
  95         double  lo;
  96 } s[INTERVALS] __aligned(16) = {
  97         0x1p+0, 0x0p+0,
  98         0x1.0163da9fb330p+0, 0x1.ab6c25335719bp-47,
  99         0x1.02c9a3e77804p+0, 0x1.07737be56527cp-47,
 100         0x1.04315e86e7f8p+0, 0x1.2f5ce3e688369p-50,
 101         0x1.059b0d315854p+0, 0x1.a1d73e2a475b4p-47,
 102         0x1.0706b29ddf6cp+0, 0x1.dc6dc403a9d88p-48,
 103         0x1.0874518759bcp+0, 0x1.01186be4bb285p-49,
 104         0x1.09e3ecac6f38p+0, 0x1.a290f03062c27p-51,
 105         0x1.0b5586cf9890p+0, 0x1.ec5317256e308p-49,
 106         0x1.0cc922b7247cp+0, 0x1.ba03db82dc49fp-47,
 107         0x1.0e3ec32d3d18p+0, 0x1.10103a1727c58p-47,
 108         0x1.0fb66affed30p+0, 0x1.af232091dd8a1p-48,
 109         0x1.11301d0125b4p+0, 0x1.0a4ebbf1aed93p-48,
 110         0x1.12abdc06c31cp+0, 0x1.7f72575a649adp-49,
 111         0x1.1429aaea92dcp+0, 0x1.fb34101943b26p-48,
 112         0x1.15a98c8a58e4p+0, 0x1.12480d573dd56p-48,
 113         0x1.172b83c7d514p+0, 0x1.d6e6fbe462876p-47,
 114         0x1.18af9388c8dcp+0, 0x1.4dddfb85cd1e1p-47,
 115         0x1.1a35beb6fcb4p+0, 0x1.a9e5b4c7b4969p-47,
 116         0x1.1bbe084045ccp+0, 0x1.39ab1e72b4428p-48,
 117         0x1.1d4873168b98p+0, 0x1.53c02dc0144c8p-47,
 118         0x1.1ed5022fcd90p+0, 0x1.cb8819ff61122p-48,
 119         0x1.2063b88628ccp+0, 0x1.63b8eeb029509p-48,
 120         0x1.21f49917ddc8p+0, 0x1.62552fd29294cp-48,
 121         0x1.2387a6e75620p+0, 0x1.c3360fd6d8e0bp-47,
 122         0x1.251ce4fb2a60p+0, 0x1.f9ac155bef4f5p-47,
 123         0x1.26b4565e27ccp+0, 0x1.d257a673281d4p-48,
 124         0x1.284dfe1f5638p+0, 0x1.2d9e2b9e07941p-53,
 125         0x1.29e9df51fdecp+0, 0x1.09612e8afad12p-47,
 126         0x1.2b87fd0dad98p+0, 0x1.ffbbd48ca71f9p-49,
 127         0x1.2d285a6e4030p+0, 0x1.680123aa6da0fp-49,
 128         0x1.2ecafa93e2f4p+0, 0x1.611ca0f45d524p-48,
 129         0x1.306fe0a31b70p+0, 0x1.52de8d5a46306p-48,
 130         0x1.32170fc4cd80p+0, 0x1.89a9ce78e1804p-47,
 131         0x1.33c08b26416cp+0, 0x1.fa64e43086cb3p-47,
 132         0x1.356c55f929fcp+0, 0x1.864a311a3b1bap-47,
 133         0x1.371a7373aa9cp+0, 0x1.54e28aa05e8a9p-49,
 134         0x1.38cae6d05d84p+0, 0x1.2c2d4e586cdf7p-47,
 135         0x1.3a7db34e59fcp+0, 0x1.b750de494cf05p-47,
 136         0x1.3c32dc313a8cp+0, 0x1.242000f9145acp-47,
 137         0x1.3dea64c12340p+0, 0x1.11ada0911f09fp-47,
 138         0x1.3fa4504ac800p+0, 0x1.ba0bf701aa418p-48,
 139         0x1.4160a21f72e0p+0, 0x1.4fc2192dc79eep-47,
 140         0x1.431f5d950a88p+0, 0x1.6dc704439410dp-48,
 141         0x1.44e086061890p+0, 0x1.68189b7a04ef8p-47,
 142         0x1.46a41ed1d004p+0, 0x1.772512f45922ap-48,
 143         0x1.486a2b5c13ccp+0, 0x1.013c1a3b69063p-48,
 144         0x1.4a32af0d7d3cp+0, 0x1.e672d8bcf46f9p-48,
 145         0x1.4bfdad5362a0p+0, 0x1.38ea1cbd7f621p-47,
 146         0x1.4dcb299fddd0p+0, 0x1.ac766dde353c2p-49,
 147         0x1.4f9b2769d2c8p+0, 0x1.35699ec5b4d50p-47,
 148         0x1.516daa2cf664p+0, 0x1.c112f52c84d82p-52,
 149         0x1.5342b569d4f8p+0, 0x1.df0a83c49d86ap-52,
 150         0x1.551a4ca5d920p+0, 0x1.d8a5d8c40486ap-49,
 151         0x1.56f4736b527cp+0, 0x1.a66ecb004764fp-48,
 152         0x1.58d12d497c7cp+0, 0x1.e9295e15b9a1ep-47,
 153         0x1.5ab07dd48540p+0, 0x1.4ac64980a8c8fp-47,
 154         0x1.5c9268a59468p+0, 0x1.b80e258dc0b4cp-47,
 155         0x1.5e76f15ad214p+0, 0x1.0dd37c9840733p-49,
 156         0x1.605e1b976dc0p+0, 0x1.160edeb25490ep-49,
 157         0x1.6247eb03a558p+0, 0x1.2c7c3e81bf4b7p-50,
 158         0x1.6434634ccc30p+0, 0x1.fc76f8714c4eep-48,
 159         0x1.662388255220p+0, 0x1.24893ecf14dc8p-47,
 160         0x1.68155d44ca94p+0, 0x1.9840e2b913dd0p-47,
 161         0x1.6a09e667f3bcp+0, 0x1.921165f626cddp-49,
 162         0x1.6c012750bda8p+0, 0x1.f76bb54cc007ap-47,
 163         0x1.6dfb23c651a0p+0, 0x1.779107165f0dep-47,
 164         0x1.6ff7df951948p+0, 0x1.e7c3f0da79f11p-51,
 165         0x1.71f75e8ec5f4p+0, 0x1.9ee91b8797785p-47,
 166         0x1.73f9a48a5814p+0, 0x1.9deae4d273456p-47,
 167         0x1.75feb564267cp+0, 0x1.17edd35467491p-49,
 168         0x1.780694fde5d0p+0, 0x1.fb0cd7014042cp-47,
 169         0x1.7a11473eb018p+0, 0x1.b5f54408fdb37p-50,
 170         0x1.7c1ed0130c10p+0, 0x1.93e2499a22c9cp-47,
 171         0x1.7e2f336cf4e4p+0, 0x1.1082e815d0abdp-47,
 172         0x1.80427543e1a0p+0, 0x1.1b60de67649a3p-48,
 173         0x1.82589994cce0p+0, 0x1.28acf88afab35p-48,
 174         0x1.8471a4623c78p+0, 0x1.667297b5cbe32p-47,
 175         0x1.868d99b4492cp+0, 0x1.640720ec85613p-47,
 176         0x1.88ac7d98a668p+0, 0x1.966530bcdf2d5p-48,
 177         0x1.8ace5422aa0cp+0, 0x1.b5ba7c55a192dp-48,
 178         0x1.8cf3216b5448p+0, 0x1.7de55439a2c39p-49,
 179         0x1.8f1ae9915770p+0, 0x1.b15cc13a2e397p-47,
 180         0x1.9145b0b91ffcp+0, 0x1.622986d1a7daep-50,
 181         0x1.93737b0cdc5cp+0, 0x1.27a280e1f92a0p-47,
 182         0x1.95a44cbc8520p+0, 0x1.dd36906d2b420p-49,
 183         0x1.97d829fde4e4p+0, 0x1.f173d241f23d1p-49,
 184         0x1.9a0f170ca078p+0, 0x1.cdd1884dc6234p-47,
 185         0x1.9c49182a3f08p+0, 0x1.01c7c46b071f3p-48,
 186         0x1.9e86319e3230p+0, 0x1.18c12653c7326p-47,
 187         0x1.a0c667b5de54p+0, 0x1.2594d6d45c656p-47,
 188         0x1.a309bec4a2d0p+0, 0x1.9ac60b8fbb86dp-47,
 189         0x1.a5503b23e254p+0, 0x1.c8b424491caf8p-48,
 190         0x1.a799e1330b34p+0, 0x1.86f2dfb2b158fp-48,
 191         0x1.a9e6b5579fd8p+0, 0x1.fa1f5921deffap-47,
 192         0x1.ac36bbfd3f34p+0, 0x1.ce06dcb351893p-47,
 193         0x1.ae89f995ad38p+0, 0x1.6af439a68bb99p-47,
 194         0x1.b0e07298db64p+0, 0x1.2c8421566fe38p-47,
 195         0x1.b33a2b84f15cp+0, 0x1.d7b5fe873decap-47,
 196         0x1.b59728de5590p+0, 0x1.cc71c40888b24p-47,
 197         0x1.b7f76f2fb5e4p+0, 0x1.baa9ec206ad4fp-50,
 198         0x1.ba5b030a1064p+0, 0x1.30819678d5eb7p-49,
 199         0x1.bcc1e904bc1cp+0, 0x1.2247ba0f45b3dp-48,
 200         0x1.bf2c25bd71e0p+0, 0x1.10811ae04a31cp-49,
 201         0x1.c199bdd85528p+0, 0x1.c2220cb12a092p-48,
 202         0x1.c40ab5fffd04p+0, 0x1.d368a6fc1078cp-47,
 203         0x1.c67f12e57d14p+0, 0x1.694426ffa41e5p-49,
 204         0x1.c8f6d9406e78p+0, 0x1.a88d65e24402ep-47,
 205         0x1.cb720dcef904p+0, 0x1.48a81e5e8f4a5p-47,
 206         0x1.cdf0b555dc3cp+0, 0x1.ce227c4ac7d63p-47,
 207         0x1.d072d4a07894p+0, 0x1.dc68791790d0bp-47,
 208         0x1.d2f87080d89cp+0, 0x1.8c56f091cc4f5p-47,
 209         0x1.d5818dcfba48p+0, 0x1.c976816bad9b8p-50,
 210         0x1.d80e316c9838p+0, 0x1.7bb84f9d04880p-48,
 211         0x1.da9e603db328p+0, 0x1.5c2300696db53p-50,
 212         0x1.dd321f301b44p+0, 0x1.025b4aef1e032p-47,
 213         0x1.dfc97337b9b4p+0, 0x1.eb968cac39ed3p-48,
 214         0x1.e264614f5a10p+0, 0x1.45093b0fd0bd7p-47,
 215         0x1.e502ee78b3fcp+0, 0x1.b139e8980a9cdp-47,
 216         0x1.e7a51fbc74c8p+0, 0x1.a5aa4594191bcp-51,
 217         0x1.ea4afa2a490cp+0, 0x1.9858f73a18f5ep-48,
 218         0x1.ecf482d8e67cp+0, 0x1.846d81897dca5p-47,
 219         0x1.efa1bee615a0p+0, 0x1.3bb8fe90d496dp-47,
 220         0x1.f252b376bba8p+0, 0x1.74e8696fc3639p-48,
 221         0x1.f50765b6e454p+0, 0x1.9d3e12dd8a18bp-54,
 222         0x1.f7bfdad9cbe0p+0, 0x1.38913b4bfe72cp-48,
 223         0x1.fa7c1819e90cp+0, 0x1.82e90a7e74b26p-48,
 224         0x1.fd3c22b8f71cp+0, 0x1.884badd25995ep-47
 225 };
 226
 227 long double
 228 expl(long double x)
 229 {
 230         union IEEEl2bits u, v;
 231         long double fn, r, r1, r2, q, t, t23, t45, twopk, twopkp10000, z;
 232         int k, n, n2;
 233         uint16_t hx, ix;
 234
 235         /* Filter out exceptional cases. */
 236         u.e = x;
 237         hx = u.xbits.expsign;
 238         ix = hx & 0x7fff;
 239         if (ix >= BIAS + 13) {          /* |x| >= 8192 or x is NaN */
 240                 if (ix == BIAS + LDBL_MAX_EXP) {
 241                         if (hx & 0x8000 && u.xbits.man == 1ULL << 63)
 242                                 return (0.0L);  /* x is -Inf */
 243                         return (x + x); /* x is +Inf, NaN or unsupported */
 244                 }
 245                 if (x > o_threshold.e)
 246                         return (huge * huge);
 247                 if (x < u_threshold.e)
 248                         return (tiny * tiny);
 249         } else if (ix <= BIAS - 34) {   /* |x| < 0x1p-33 */
 250                                         /* includes pseudo-denormals */
 251                 if (huge + x > 1.0L)    /* trigger inexact iff x != 0 */
 252                         return (1.0L + x);
 253         }
 254
 255         ENTERI();
 256
 257         /* Reduce x to (k*ln2 + midpoint[n2] + r1 + r2). */
 258         /* Use a specialized rint() to get fn.  Assume round-to-nearest. */
 259         fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
 260         r = x - fn * L1 - fn * L2;      /* r = r1 + r2 done independently. */
 261 #if defined(HAVE_EFFICIENT_IRINTL)
 262         n  = irintl(fn);
 263 #elif defined(HAVE_EFFICIENT_IRINT)
 264         n  = irint(fn);
 265 #else
 266         n  = (int)fn;
 267 #endif
 268         n2 = (unsigned)n % INTERVALS;           /* Tang's j. */
 269         k = (n - n2) / INTERVALS;
 270         r1 = x - fn * L1;
 271         r2 = -fn * L2;
 272
 273         /* Prepare scale factors. */
 274         v.xbits.man = 1ULL << 63;
 275         if (k >= LDBL_MIN_EXP) {
 276                 v.xbits.expsign = BIAS + k;
 277                 twopk = v.e;
 278         } else {
 279                 v.xbits.expsign = BIAS + k + 10000;
 280                 twopkp10000 = v.e;
 281         }
 282
 283         /* Evaluate expl(midpoint[n2] + r1 + r2) = s[n2] * expl(r1 + r2). */
 284         /* Here q = q(r), not q(r1), since r1 is lopped like L1. */
 285         t45 = r * P5 + P4;
 286         z = r * r;
 287         t23 = r * P3 + P2;
 288         q = r2 + z * t23 + z * z * t45 + z * z * z * P6;
 289         t = (long double)s[n2].lo + s[n2].hi;
 290         t = s[n2].lo + t * (q + r1) + s[n2].hi;
 291
 292         /* Scale by 2**k. */
 293         if (k >= LDBL_MIN_EXP) {
 294                 if (k == LDBL_MAX_EXP)
 295                         RETURNI(t * 2.0L * 0x1p16383L);
 296                 RETURNI(t * twopk);
 297         } else {
 298                 RETURNI(t * twopkp10000 * twom10000);
 299         }
 300 }