usr/src/lib/libm/common/C/exp.c

   1 /*
   2  * CDDL HEADER START
   3  *
   4  * The contents of this file are subject to the terms of the
   5  * Common Development and Distribution License (the "License").
   6  * You may not use this file except in compliance with the License.
   7  *
   8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
   9  * or http://www.opensolaris.org/os/licensing.
  10  * See the License for the specific language governing permissions
  11  * and limitations under the License.
  12  *
  13  * When distributing Covered Code, include this CDDL HEADER in each
  14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
  15  * If applicable, add the following below this CDDL HEADER, with the
  16  * fields enclosed by brackets "[]" replaced with your own identifying
  17  * information: Portions Copyright [yyyy] [name of copyright owner]
  18  *
  19  * CDDL HEADER END
  20  */
  21 /*
  22  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  23  */
  24 /*
  25  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  26  * Use is subject to license terms.
  27  */
  28
  29 #pragma weak __exp = exp
  30
  31 /*
  32  * exp(x)
  33  * Hybrid algorithm of Peter Tang's Table driven method (for large
  34  * arguments) and an accurate table (for small arguments).
  35  * Written by K.C. Ng, November 1988.
  36  * Method (large arguments):
  37  *      1. Argument Reduction: given the input x, find r and integer k
  38  *         and j such that
  39  *                   x = (k+j/32)*(ln2) + r,  |r| <= (1/64)*ln2
  40  *
  41  *      2. exp(x) = 2^k * (2^(j/32) + 2^(j/32)*expm1(r))
  42  *         a. expm1(r) is approximated by a polynomial:
  43  *            expm1(r) ~ r + t1*r^2 + t2*r^3 + ... + t5*r^6
  44  *            Here t1 = 1/2 exactly.
  45  *         b. 2^(j/32) is represented to twice double precision
  46  *            as TBL[2j]+TBL[2j+1].
  47  *
  48  * Note: If divide were fast enough, we could use another approximation
  49  *       in 2.a:
  50  *            expm1(r) ~ (2r)/(2-R), R = r - r^2*(t1 + t2*r^2)
  51  *            (for the same t1 and t2 as above)
  52  *
  53  * Special cases:
  54  *      exp(INF) is INF, exp(NaN) is NaN;
  55  *      exp(-INF)=  0;
  56  *      for finite argument, only exp(0)=1 is exact.
  57  *
  58  * Accuracy:
  59  *      According to an error analysis, the error is always less than
  60  *      an ulp (unit in the last place).  The largest errors observed
  61  *      are less than 0.55 ulp for normal results and less than 0.75 ulp
  62  *      for subnormal results.
  63  *
  64  * Misc. info.
  65  *      For IEEE double
  66  *              if x >  7.09782712893383973096e+02 then exp(x) overflow
  67  *              if x < -7.45133219101941108420e+02 then exp(x) underflow
  68  */
  69
  70 #include "libm.h"
  71
  72 static const double TBL[] = {
  73         1.00000000000000000000e+00,  0.00000000000000000000e+00,
  74         1.02189714865411662714e+00,  5.10922502897344389359e-17,
  75         1.04427378242741375480e+00,  8.55188970553796365958e-17,
  76         1.06714040067682369717e+00, -7.89985396684158212226e-17,
  77         1.09050773266525768967e+00, -3.04678207981247114697e-17,
  78         1.11438674259589243221e+00,  1.04102784568455709549e-16,
  79         1.13878863475669156458e+00,  8.91281267602540777782e-17,
  80         1.16372485877757747552e+00,  3.82920483692409349872e-17,
  81         1.18920711500272102690e+00,  3.98201523146564611098e-17,
  82         1.21524735998046895524e+00, -7.71263069268148813091e-17,
  83         1.24185781207348400201e+00,  4.65802759183693679123e-17,
  84         1.26905095719173321989e+00,  2.66793213134218609523e-18,
  85         1.29683955465100964055e+00,  2.53825027948883149593e-17,
  86         1.32523664315974132322e+00, -2.85873121003886075697e-17,
  87         1.35425554693689265129e+00,  7.70094837980298946162e-17,
  88         1.38390988196383202258e+00, -6.77051165879478628716e-17,
  89         1.41421356237309514547e+00, -9.66729331345291345105e-17,
  90         1.44518080697704665027e+00, -3.02375813499398731940e-17,
  91         1.47682614593949934623e+00, -3.48399455689279579579e-17,
  92         1.50916442759342284141e+00, -1.01645532775429503911e-16,
  93         1.54221082540794074411e+00,  7.94983480969762085616e-17,
  94         1.57598084510788649659e+00, -1.01369164712783039808e-17,
  95         1.61049033194925428347e+00,  2.47071925697978878522e-17,
  96         1.64575547815396494578e+00, -1.01256799136747726038e-16,
  97         1.68179283050742900407e+00,  8.19901002058149652013e-17,
  98         1.71861929812247793414e+00, -1.85138041826311098821e-17,
  99         1.75625216037329945351e+00,  2.96014069544887330703e-17,
 100         1.79470907500310716820e+00,  1.82274584279120867698e-17,
 101         1.83400808640934243066e+00,  3.28310722424562658722e-17,
 102         1.87416763411029996256e+00, -6.12276341300414256164e-17,
 103         1.91520656139714740007e+00, -1.06199460561959626376e-16,
 104         1.95714412417540017941e+00,  8.96076779103666776760e-17,
 105 };
 106
 107 /*
 108  * For i = 0, ..., 66,
 109  *   TBL2[2*i] is a double precision number near (i+1)*2^-6, and
 110  *   TBL2[2*i+1] = exp(TBL2[2*i]) to within a relative error less
 111  *   than 2^-60.
 112  *
 113  * For i = 67, ..., 133,
 114  *   TBL2[2*i] is a double precision number near -(i+1)*2^-6, and
 115  *   TBL2[2*i+1] = exp(TBL2[2*i]) to within a relative error less
 116  *   than 2^-60.
 117  */
 118 static const double TBL2[] = {
 119         1.56249999999984491572e-02, 1.01574770858668417262e+00,
 120         3.12499999999998716305e-02, 1.03174340749910253834e+00,
 121         4.68750000000011102230e-02, 1.04799100201663386578e+00,
 122         6.24999999999990632493e-02, 1.06449445891785843266e+00,
 123         7.81249999999999444888e-02, 1.08125780744903954300e+00,
 124         9.37500000000013322676e-02, 1.09828514030782731226e+00,
 125         1.09375000000001346145e-01, 1.11558061464248226002e+00,
 126         1.24999999999999417133e-01, 1.13314845306682565607e+00,
 127         1.40624999999995337063e-01, 1.15099294469117108264e+00,
 128         1.56249999999996141975e-01, 1.16911844616949989195e+00,
 129         1.71874999999992894573e-01, 1.18752938276309216725e+00,
 130         1.87500000000000888178e-01, 1.20623024942098178158e+00,
 131         2.03124999999361649516e-01, 1.22522561187652545556e+00,
 132         2.18750000000000416334e-01, 1.24452010776609567344e+00,
 133         2.34375000000003524958e-01, 1.26411844775347081971e+00,
 134         2.50000000000006328271e-01, 1.28402541668774961003e+00,
 135         2.65624999999982791543e-01, 1.30424587476761533189e+00,
 136         2.81249999999993727240e-01, 1.32478475872885725906e+00,
 137         2.96875000000003275158e-01, 1.34564708304941493822e+00,
 138         3.12500000000002886580e-01, 1.36683794117380030819e+00,
 139         3.28124999999993394173e-01, 1.38836250675661765364e+00,
 140         3.43749999999998612221e-01, 1.41022603492570874906e+00,
 141         3.59374999999992450483e-01, 1.43243386356506730017e+00,
 142         3.74999999999991395772e-01, 1.45499141461818881638e+00,
 143         3.90624999999997613020e-01, 1.47790419541173490003e+00,
 144         4.06249999999991895372e-01, 1.50117780000011058483e+00,
 145         4.21874999999996613820e-01, 1.52481791053132154090e+00,
 146         4.37500000000004607426e-01, 1.54883029863414023453e+00,
 147         4.53125000000004274359e-01, 1.57322082682725961078e+00,
 148         4.68750000000008326673e-01, 1.59799544995064657371e+00,
 149         4.84374999999985456078e-01, 1.62316021661928200359e+00,
 150         4.99999999999997335465e-01, 1.64872127070012375327e+00,
 151         5.15625000000000222045e-01, 1.67468485281178436352e+00,
 152         5.31250000000003441691e-01, 1.70105730184840653330e+00,
 153         5.46874999999999111822e-01, 1.72784505652716169344e+00,
 154         5.62499999999999333866e-01, 1.75505465696029738787e+00,
 155         5.78124999999993338662e-01, 1.78269274625180318417e+00,
 156         5.93749999999999666933e-01, 1.81076607211938656050e+00,
 157         6.09375000000003441691e-01, 1.83928148854178719063e+00,
 158         6.24999999999995559108e-01, 1.86824595743221411048e+00,
 159         6.40625000000009103829e-01, 1.89766655033813602671e+00,
 160         6.56249999999993782751e-01, 1.92755045016753268072e+00,
 161         6.71875000000002109424e-01, 1.95790495294292221651e+00,
 162         6.87499999999992450483e-01, 1.98873746958227681780e+00,
 163         7.03125000000004996004e-01, 2.02005552770870666635e+00,
 164         7.18750000000007105427e-01, 2.05186677348799140219e+00,
 165         7.34375000000008770762e-01, 2.08417897349558689513e+00,
 166         7.49999999999983901766e-01, 2.11700001661264058939e+00,
 167         7.65624999999997002398e-01, 2.15033791595229351046e+00,
 168         7.81250000000005884182e-01, 2.18420081081563077774e+00,
 169         7.96874999999991451283e-01, 2.21859696867912603579e+00,
 170         8.12500000000000000000e-01, 2.25353478721320854561e+00,
 171         8.28125000000008215650e-01, 2.28902279633221983346e+00,
 172         8.43749999999997890576e-01, 2.32506966027711614586e+00,
 173         8.59374999999999444888e-01, 2.36168417973090827289e+00,
 174         8.75000000000003219647e-01, 2.39887529396710563745e+00,
 175         8.90625000000013433699e-01, 2.43665208303232461162e+00,
 176         9.06249999999980571097e-01, 2.47502376996297712708e+00,
 177         9.21874999999984456878e-01, 2.51399972303748420188e+00,
 178         9.37500000000001887379e-01, 2.55358945806293169412e+00,
 179         9.53125000000003330669e-01, 2.59380264069854327147e+00,
 180         9.68749999999989119814e-01, 2.63464908881560244680e+00,
 181         9.84374999999997890576e-01, 2.67613877489447116176e+00,
 182         1.00000000000001154632e+00, 2.71828182845907662113e+00,
 183         1.01562499999999333866e+00, 2.76108853855008318234e+00,
 184         1.03124999999995980993e+00, 2.80456935623711389738e+00,
 185         1.04687499999999933387e+00, 2.84873489717039740654e+00,
 186         -1.56249999999999514277e-02, 9.84496437005408453480e-01,
 187         -3.12499999999955972718e-02, 9.69233234476348348707e-01,
 188         -4.68749999999993824384e-02, 9.54206665969188905230e-01,
 189         -6.24999999999976130205e-02, 9.39413062813478028090e-01,
 190         -7.81249999999989314103e-02, 9.24848813216205822840e-01,
 191         -9.37499999999995975442e-02, 9.10510361380034494161e-01,
 192         -1.09374999999998584466e-01, 8.96394206635151680196e-01,
 193         -1.24999999999998556710e-01, 8.82496902584596676355e-01,
 194         -1.40624999999999361622e-01, 8.68815056262843721235e-01,
 195         -1.56249999999999111822e-01, 8.55345327307423297647e-01,
 196         -1.71874999999924144012e-01, 8.42084427143446223596e-01,
 197         -1.87499999999996752598e-01, 8.29029118180403035154e-01,
 198         -2.03124999999988037347e-01, 8.16176213022349550386e-01,
 199         -2.18749999999995947686e-01, 8.03522573689063990265e-01,
 200         -2.34374999999996419531e-01, 7.91065110850298847112e-01,
 201         -2.49999999999996280753e-01, 7.78800783071407765057e-01,
 202         -2.65624999999999888978e-01, 7.66726596070820165529e-01,
 203         -2.81249999999989397370e-01, 7.54839601989015340777e-01,
 204         -2.96874999999996114219e-01, 7.43136898668761203268e-01,
 205         -3.12499999999999555911e-01, 7.31615628946642115871e-01,
 206         -3.28124999999993782751e-01, 7.20272979955444259126e-01,
 207         -3.43749999999997946087e-01, 7.09106182437399867879e-01,
 208         -3.59374999999994337863e-01, 6.98112510068129799023e-01,
 209         -3.74999999999994615418e-01, 6.87289278790975899369e-01,
 210         -3.90624999999999000799e-01, 6.76633846161729612945e-01,
 211         -4.06249999999947264406e-01, 6.66143610703522903727e-01,
 212         -4.21874999999988453681e-01, 6.55816011271509125002e-01,
 213         -4.37499999999999111822e-01, 6.45648526427892610613e-01,
 214         -4.53124999999999278355e-01, 6.35638673826052436056e-01,
 215         -4.68749999999999278355e-01, 6.25784009604591573428e-01,
 216         -4.84374999999992894573e-01, 6.16082127790682609891e-01,
 217         -4.99999999999998168132e-01, 6.06530659712634534486e-01,
 218         -5.15625000000000000000e-01, 5.97127273421627413619e-01,
 219         -5.31249999999989785948e-01, 5.87869673122352498496e-01,
 220         -5.46874999999972688514e-01, 5.78755598612500032907e-01,
 221         -5.62500000000000000000e-01, 5.69782824730923009859e-01,
 222         -5.78124999999992339461e-01, 5.60949160814475100700e-01,
 223         -5.93749999999948707696e-01, 5.52252450163048691500e-01,
 224         -6.09374999999552580121e-01, 5.43690569513243682209e-01,
 225         -6.24999999999984789945e-01, 5.35261428518998383375e-01,
 226         -6.40624999999983457677e-01, 5.26962969243379708573e-01,
 227         -6.56249999999998334665e-01, 5.18793165653890220312e-01,
 228         -6.71874999999943378626e-01, 5.10750023129039609771e-01,
 229         -6.87499999999997002398e-01, 5.02831577970942467104e-01,
 230         -7.03124999999991118216e-01, 4.95035896926202978463e-01,
 231         -7.18749999999991340260e-01, 4.87361076713623331269e-01,
 232         -7.34374999999985678123e-01, 4.79805243559684402310e-01,
 233         -7.49999999999997335465e-01, 4.72366552741015965911e-01,
 234         -7.65624999999993782751e-01, 4.65043188134059204408e-01,
 235         -7.81249999999863220523e-01, 4.57833361771676883301e-01,
 236         -7.96874999999998112621e-01, 4.50735313406363247157e-01,
 237         -8.12499999999990119015e-01, 4.43747310081084256339e-01,
 238         -8.28124999999996003197e-01, 4.36867645705559026759e-01,
 239         -8.43749999999988120614e-01, 4.30094640640067360504e-01,
 240         -8.59374999999994115818e-01, 4.23426641285265303871e-01,
 241         -8.74999999999977129406e-01, 4.16862019678517936594e-01,
 242         -8.90624999999983346655e-01, 4.10399173096376801428e-01,
 243         -9.06249999999991784350e-01, 4.04036523663345414903e-01,
 244         -9.21874999999994004796e-01, 3.97772517966614058693e-01,
 245         -9.37499999999994337863e-01, 3.91605626676801210628e-01,
 246         -9.53124999999999444888e-01, 3.85534344174578935682e-01,
 247         -9.68749999999986677324e-01, 3.79557188183094640355e-01,
 248         -9.84374999999992339461e-01, 3.73672699406045860648e-01,
 249         -9.99999999999995892175e-01, 3.67879441171443832825e-01,
 250         -1.01562499999994315658e+00, 3.62175999080846300338e-01,
 251         -1.03124999999991096011e+00, 3.56560980663978732697e-01,
 252         -1.04687499999999067413e+00, 3.51033015038813400732e-01,
 253 };
 254
 255 static const double C[] = {
 256         0.5,
 257         4.61662413084468283841e+01,     /* 0x40471547, 0x652b82fe */
 258         2.16608493865351192653e-02,     /* 0x3f962e42, 0xfee00000 */
 259         5.96317165397058656257e-12,     /* 0x3d9a39ef, 0x35793c76 */
 260         1.6666666666526086527e-1,       /* 3fc5555555548f7c */
 261         4.1666666666226079285e-2,       /* 3fa5555555545d4e */
 262         8.3333679843421958056e-3,       /* 3f811115b7aa905e */
 263         1.3888949086377719040e-3,       /* 3f56c1728d739765 */
 264         1.0,
 265         0.0,
 266         7.09782712893383973096e+02,     /* 0x40862E42, 0xFEFA39EF */
 267         7.45133219101941108420e+02,     /* 0x40874910, 0xD52D3051 */
 268         5.55111512312578270212e-17,     /* 0x3c900000, 0x00000000 */
 269 };
 270
 271 #define half            C[0]
 272 #define invln2_32       C[1]
 273 #define ln2_32hi        C[2]
 274 #define ln2_32lo        C[3]
 275 #define t2              C[4]
 276 #define t3              C[5]
 277 #define t4              C[6]
 278 #define t5              C[7]
 279 #define one             C[8]
 280 #define zero            C[9]
 281 #define threshold1      C[10]
 282 #define threshold2      C[11]
 283 #define twom54          C[12]
 284
 285 double
 286 exp(double x) {
 287         double  y, z, t;
 288         int     hx, ix, k, j, m;
 289
 290         ix = ((int *)&x)[HIWORD];
 291         hx = ix & ~0x80000000;
 292
 293         if (hx < 0x3ff0a2b2) {  /* |x| < 3/2 ln 2 */
 294                 if (hx < 0x3f862e42) {  /* |x| < 1/64 ln 2 */
 295                         if (hx < 0x3ed00000) {  /* |x| < 2^-18 */
 296                                 volatile int dummy __unused;
 297
 298                                 dummy = (int)x; /* raise inexact if x != 0 */
 299                                 if (hx < 0x3e300000)
 300                                         return (one + x);
 301                                 return (one + x * (one + half * x));
 302                         }
 303                         t = x * x;
 304                         y = x + (t * (half + x * t2) +
 305                             (t * t) * (t3 + x * t4 + t * t5));
 306                         return (one + y);
 307                 }
 308
 309                 /* find the multiple of 2^-6 nearest x */
 310                 k = hx >> 20;
 311                 j = (0x00100000 | (hx & 0x000fffff)) >> (0x40c - k);
 312                 j = (j - 1) & ~1;
 313                 if (ix < 0)
 314                         j += 134;
 315                 z = x - TBL2[j];
 316                 t = z * z;
 317                 y = z + (t * (half + z * t2) +
 318                     (t * t) * (t3 + z * t4 + t * t5));
 319                 return (TBL2[j+1] + TBL2[j+1] * y);
 320         }
 321
 322         if (hx >= 0x40862e42) { /* x is large, infinite, or nan */
 323                 if (hx >= 0x7ff00000) {
 324                         if (ix == 0xfff00000 && ((int *)&x)[LOWORD] == 0)
 325                                 return (zero);
 326                         return (x * x);
 327                 }
 328                 if (x > threshold1)
 329                         return (_SVID_libm_err(x, x, 6));
 330                 if (-x > threshold2)
 331                         return (_SVID_libm_err(x, x, 7));
 332         }
 333
 334         t = invln2_32 * x;
 335         if (ix < 0)
 336                 t -= half;
 337         else
 338                 t += half;
 339         k = (int)t;
 340         j = (k & 0x1f) << 1;
 341         m = k >> 5;
 342         z = (x - k * ln2_32hi) - k * ln2_32lo;
 343
 344         /* z is now in primary range */
 345         t = z * z;
 346         y = z + (t * (half + z * t2) + (t * t) * (t3 + z * t4 + t * t5));
 347         y = TBL[j] + (TBL[j+1] + TBL[j] * y);
 348         if (m < -1021) {
 349                 ((int *)&y)[HIWORD] += (m + 54) << 20;
 350                 return (twom54 * y);
 351         }
 352         ((int *)&y)[HIWORD] += m << 20;
 353         return (y);
 354 }