usr/src/lib/libm/common/R/expf.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 __expf = expf
  30
  31 /* INDENT OFF */
  32 /*
  33  * float expf(float x);
  34  * Code by K.C. Ng for SUN 5.0 libmopt
  35  * 11/5/99
  36  * Method :
  37  *      1. For |x| >= 2^7, either underflow/overflow.
  38  *         More precisely:
  39  *              x > 88.722839355...(0x42B17218) => overflow;
  40  *              x < -103.97207642..(0xc2CFF1B4) => underflow.
  41  *      2. For |x| <  2^-6, use polynomail
  42  *              exp(x) = 1 + x + p1*x^2 + p2*x^3
  43  *      3. Otherwise, write |x|=(1+r)*2^n, where 0<=r<1.
  44  *         Let t = 2^n * (1+r) .... x > 0;
  45  *             t = 2^n * (1-r) .... x < 0. (x= -2**(n+1)+t)
  46  *         Since -6 <= n <= 6, we may break t into
  47  *         six 6-bits chunks:
  48  *                    -5     -11     -17     -23     -29
  49  *         t=j *2+j *2  +j *2   +j *2   +j *2   +j *2
  50  *            1    2      3       4       5       6
  51  *
  52  *         where 0 <= j  < 64 for i = 1,...,6.
  53  *                     i
  54  *         Note that since t has only 24 significant bits,
  55  *         either j  or j  must be 0.
  56  *                 1     6
  57  *                                             7-6i
  58  *         One may define j  by   (int) ( t * 2     ) mod 64
  59  *                         i
  60  *         mathematically. In actual implementation, they can
  61  *         be obtained by manipulating the exponent and
  62  *         mantissa bits as follow:
  63  *              Let ix = (HEX(x)&0x007fffff)|0x00800000.
  64  *              If n>=0, let ix=ix<<n, then j =0 and
  65  *                                           6
  66  *                  j  = ix>>(30-6i)) mod 64  ...i=1,...,5
  67  *                   i
  68  *              Otherwise, let ix=ix<<(j+6), then j = 0 and
  69  *                                                 1
  70  *                  j  = ix>>(36-6i)) mod 64  ...i=2,...,6
  71  *                   i
  72  *
  73  *      4. Compute exp(t) by table look-up method.
  74  *         Precompute ET[k] = exp(j*2^(7-6i)), k=j+64*(6-i).
  75  *         Then
  76  *         exp(t) = ET[j +320]*ET[j +256]*ET[j +192]*
  77  *                      1          2          3
  78  *
  79  *                  ET[j +128]*ET[j +64]*ET[j ]
  80  *                      4          5         6
  81  *
  82  *                                n+1
  83  *      5. If x < 0, return exp(-2   )* exp(t). Note that
  84  *         -6 <= n <= 6. Let k = n - 6, then we can
  85  *         precompute
  86  *                       k-5          n+1
  87  *         EN[k] = exp(-2   ) = exp(-2   ) for k=0,1,...,12.
  88  *
  89  *
  90  * Special cases:
  91  *      exp(INF) is INF, exp(NaN) is NaN;
  92  *      exp(-INF) = 0;
  93  *      for finite argument, only exp(0) = 1 is exact.
  94  *
  95  * Accuracy:
  96  *      All calculations are done in double precision except for
  97  *      the case |x| < 2^-6.  When |x| < 2^-6, the error is less
  98  *      than 0.55 ulp.  When |x| >= 2^-6 and the result is normal,
  99  *      the error is less than 0.51 ulp.  When FDTOS_TRAPS_... is
 100  *      defined and the result is subnormal, the error can be as
 101  *      large as 0.75 ulp.
 102  */
 103 /* INDENT ON */
 104
 105 #include "libm.h"
 106
 107 /*
 108  * ET[k] = exp(j*2^(7-6i)) , where j = k mod 64, i = k/64
 109  */
 110 static const double ET[] = {
 111         1.00000000000000000000e+00, 1.00000000186264514923e+00,
 112         1.00000000372529029846e+00, 1.00000000558793544769e+00,
 113         1.00000000745058059692e+00, 1.00000000931322574615e+00,
 114         1.00000001117587089539e+00, 1.00000001303851604462e+00,
 115         1.00000001490116119385e+00, 1.00000001676380656512e+00,
 116         1.00000001862645171435e+00, 1.00000002048909686359e+00,
 117         1.00000002235174201282e+00, 1.00000002421438716205e+00,
 118         1.00000002607703253332e+00, 1.00000002793967768255e+00,
 119         1.00000002980232283178e+00, 1.00000003166496798102e+00,
 120         1.00000003352761335229e+00, 1.00000003539025850152e+00,
 121         1.00000003725290365075e+00, 1.00000003911554879998e+00,
 122         1.00000004097819417126e+00, 1.00000004284083932049e+00,
 123         1.00000004470348446972e+00, 1.00000004656612984100e+00,
 124         1.00000004842877499023e+00, 1.00000005029142036150e+00,
 125         1.00000005215406551073e+00, 1.00000005401671088201e+00,
 126         1.00000005587935603124e+00, 1.00000005774200140252e+00,
 127         1.00000005960464655175e+00, 1.00000006146729192302e+00,
 128         1.00000006332993707225e+00, 1.00000006519258244353e+00,
 129         1.00000006705522759276e+00, 1.00000006891787296404e+00,
 130         1.00000007078051811327e+00, 1.00000007264316348454e+00,
 131         1.00000007450580863377e+00, 1.00000007636845400505e+00,
 132         1.00000007823109937632e+00, 1.00000008009374452556e+00,
 133         1.00000008195638989683e+00, 1.00000008381903526811e+00,
 134         1.00000008568168063938e+00, 1.00000008754432578861e+00,
 135         1.00000008940697115989e+00, 1.00000009126961653116e+00,
 136         1.00000009313226190244e+00, 1.00000009499490705167e+00,
 137         1.00000009685755242295e+00, 1.00000009872019779422e+00,
 138         1.00000010058284316550e+00, 1.00000010244548853677e+00,
 139         1.00000010430813368600e+00, 1.00000010617077905728e+00,
 140         1.00000010803342442856e+00, 1.00000010989606979983e+00,
 141         1.00000011175871517111e+00, 1.00000011362136054238e+00,
 142         1.00000011548400591366e+00, 1.00000011734665128493e+00,
 143         1.00000000000000000000e+00, 1.00000011920929665621e+00,
 144         1.00000023841860752327e+00, 1.00000035762793260119e+00,
 145         1.00000047683727188996e+00, 1.00000059604662538959e+00,
 146         1.00000071525599310007e+00, 1.00000083446537502141e+00,
 147         1.00000095367477115360e+00, 1.00000107288418149665e+00,
 148         1.00000119209360605055e+00, 1.00000131130304481530e+00,
 149         1.00000143051249779091e+00, 1.00000154972196497738e+00,
 150         1.00000166893144637470e+00, 1.00000178814094198287e+00,
 151         1.00000190735045180190e+00, 1.00000202655997583179e+00,
 152         1.00000214576951407253e+00, 1.00000226497906652412e+00,
 153         1.00000238418863318657e+00, 1.00000250339821405987e+00,
 154         1.00000262260780914403e+00, 1.00000274181741843904e+00,
 155         1.00000286102704194491e+00, 1.00000298023667966163e+00,
 156         1.00000309944633158921e+00, 1.00000321865599772764e+00,
 157         1.00000333786567807692e+00, 1.00000345707537263706e+00,
 158         1.00000357628508140806e+00, 1.00000369549480438991e+00,
 159         1.00000381470454158261e+00, 1.00000393391429298617e+00,
 160         1.00000405312405860059e+00, 1.00000417233383842586e+00,
 161         1.00000429154363246198e+00, 1.00000441075344070896e+00,
 162         1.00000452996326316679e+00, 1.00000464917309983548e+00,
 163         1.00000476838295071502e+00, 1.00000488759281580542e+00,
 164         1.00000500680269510667e+00, 1.00000512601258861878e+00,
 165         1.00000524522249634174e+00, 1.00000536443241827556e+00,
 166         1.00000548364235442023e+00, 1.00000560285230477575e+00,
 167         1.00000572206226934213e+00, 1.00000584127224811937e+00,
 168         1.00000596048224110746e+00, 1.00000607969224830640e+00,
 169         1.00000619890226971620e+00, 1.00000631811230533685e+00,
 170         1.00000643732235516836e+00, 1.00000655653241921073e+00,
 171         1.00000667574249746394e+00, 1.00000679495258992802e+00,
 172         1.00000691416269660294e+00, 1.00000703337281748873e+00,
 173         1.00000715258295258536e+00, 1.00000727179310189285e+00,
 174         1.00000739100326541120e+00, 1.00000751021344314040e+00,
 175         1.00000000000000000000e+00, 1.00000762942363508046e+00,
 176         1.00001525890547848796e+00, 1.00002288844553022251e+00,
 177         1.00003051804379095024e+00, 1.00003814770026133729e+00,
 178         1.00004577741494138365e+00, 1.00005340718783175546e+00,
 179         1.00006103701893311886e+00, 1.00006866690824547383e+00,
 180         1.00007629685576948653e+00, 1.00008392686150582307e+00,
 181         1.00009155692545448346e+00, 1.00009918704761613384e+00,
 182         1.00010681722799144033e+00, 1.00011444746658040295e+00,
 183         1.00012207776338368781e+00, 1.00012970811840196106e+00,
 184         1.00013733853163522269e+00, 1.00014496900308413885e+00,
 185         1.00015259953274937565e+00, 1.00016023012063093311e+00,
 186         1.00016786076672947736e+00, 1.00017549147104567453e+00,
 187         1.00018312223357952462e+00, 1.00019075305433191581e+00,
 188         1.00019838393330284809e+00, 1.00020601487049298761e+00,
 189         1.00021364586590300050e+00, 1.00022127691953288675e+00,
 190         1.00022890803138353455e+00, 1.00023653920145494389e+00,
 191         1.00024417042974778091e+00, 1.00025180171626271175e+00,
 192         1.00025943306099973640e+00, 1.00026706446395974304e+00,
 193         1.00027469592514273167e+00, 1.00028232744454959047e+00,
 194         1.00028995902218031944e+00, 1.00029759065803558471e+00,
 195         1.00030522235211605242e+00, 1.00031285410442172257e+00,
 196         1.00032048591495348333e+00, 1.00032811778371155675e+00,
 197         1.00033574971069616488e+00, 1.00034338169590819589e+00,
 198         1.00035101373934764979e+00, 1.00035864584101541475e+00,
 199         1.00036627800091149076e+00, 1.00037391021903676602e+00,
 200         1.00038154249539146257e+00, 1.00038917482997580244e+00,
 201         1.00039680722279067382e+00, 1.00040443967383629875e+00,
 202         1.00041207218311289928e+00, 1.00041970475062136359e+00,
 203         1.00042733737636191371e+00, 1.00043497006033499375e+00,
 204         1.00044260280254104778e+00, 1.00045023560298029786e+00,
 205         1.00045786846165363215e+00, 1.00046550137856127272e+00,
 206         1.00047313435370366363e+00, 1.00048076738708124900e+00,
 207         1.00000000000000000000e+00, 1.00048840047869447289e+00,
 208         1.00097703949241645383e+00, 1.00146591715766675179e+00,
 209         1.00195503359100279717e+00, 1.00244438890903908579e+00,
 210         1.00293398322844673487e+00, 1.00342381666595459322e+00,
 211         1.00391388933834746489e+00, 1.00440420136246855165e+00,
 212         1.00489475285521656645e+00, 1.00538554393354861993e+00,
 213         1.00587657471447822211e+00, 1.00636784531507639251e+00,
 214         1.00685935585247099411e+00, 1.00735110644384739942e+00,
 215         1.00784309720644804642e+00, 1.00833532825757243856e+00,
 216         1.00882779971457803292e+00, 1.00932051169487890796e+00,
 217         1.00981346431594687374e+00, 1.01030665769531102782e+00,
 218         1.01080009195055753324e+00, 1.01129376719933050666e+00,
 219         1.01178768355933157430e+00, 1.01228184114831898377e+00,
 220         1.01277624008410960244e+00, 1.01327088048457714109e+00,
 221         1.01376576246765282008e+00, 1.01426088615132625748e+00,
 222         1.01475625165364347069e+00, 1.01525185909270931894e+00,
 223         1.01574770858668572693e+00, 1.01624380025379235093e+00,
 224         1.01674013421230657883e+00, 1.01723671058056375216e+00,
 225         1.01773352947695694404e+00, 1.01823059101993673714e+00,
 226         1.01872789532801233392e+00, 1.01922544251975000229e+00,
 227         1.01972323271377418585e+00, 1.02022126602876750390e+00,
 228         1.02071954258347008526e+00, 1.02121806249668067856e+00,
 229         1.02171682588725554197e+00, 1.02221583287410910934e+00,
 230         1.02271508357621376817e+00, 1.02321457811260052573e+00,
 231         1.02371431660235789884e+00, 1.02421429916463280207e+00,
 232         1.02471452591863054771e+00, 1.02521499698361440167e+00,
 233         1.02571571247890602763e+00, 1.02621667252388526492e+00,
 234         1.02671787723799012859e+00, 1.02721932674071725344e+00,
 235         1.02772102115162167202e+00, 1.02822296059031659254e+00,
 236         1.02872514517647339893e+00, 1.02922757502982276101e+00,
 237         1.02973025027015285815e+00, 1.03023317101731093359e+00,
 238         1.03073633739120262831e+00, 1.03123974951179242510e+00,
 239         1.00000000000000000000e+00, 1.03174340749910276038e+00,
 240         1.06449445891785954288e+00, 1.09828514030782575794e+00,
 241         1.13314845306682632220e+00, 1.16911844616950433284e+00,
 242         1.20623024942098067136e+00, 1.24452010776609522935e+00,
 243         1.28402541668774139438e+00, 1.32478475872886569675e+00,
 244         1.36683794117379631139e+00, 1.41022603492571074746e+00,
 245         1.45499141461820125087e+00, 1.50117780000012279729e+00,
 246         1.54883029863413312910e+00, 1.59799544995063325104e+00,
 247         1.64872127070012819416e+00, 1.70105730184840076014e+00,
 248         1.75505465696029849809e+00, 1.81076607211938722664e+00,
 249         1.86824595743222232613e+00, 1.92755045016754467113e+00,
 250         1.98873746958229191684e+00, 2.05186677348797674725e+00,
 251         2.11700001661267478426e+00, 2.18420081081561789915e+00,
 252         2.25353478721320854561e+00, 2.32506966027712103084e+00,
 253         2.39887529396709808793e+00, 2.47502376996302508871e+00,
 254         2.55358945806292680913e+00, 2.63464908881563086851e+00,
 255         2.71828182845904553488e+00, 2.80456935623722669604e+00,
 256         2.89359594417176113623e+00, 2.98544853936535581340e+00,
 257         3.08021684891803104733e+00, 3.17799342753883840018e+00,
 258         3.27887376793867346692e+00, 3.38295639409246895468e+00,
 259         3.49034295746184142217e+00, 3.60113833627217561073e+00,
 260         3.71545073794110392029e+00, 3.83339180475841034834e+00,
 261         3.95507672292057721464e+00, 4.08062433502646015882e+00,
 262         4.21015725614395996956e+00, 4.34380199356104235164e+00,
 263         4.48168907033806451778e+00, 4.62395315278208052234e+00,
 264         4.77073318196760265408e+00, 4.92217250943229078786e+00,
 265         5.07841903718008147450e+00, 5.23962536212848917216e+00,
 266         5.40594892514116676097e+00, 5.57755216479125959239e+00,
 267         5.75460267600573072144e+00, 5.93727337374560715233e+00,
 268         6.12574266188198635064e+00, 6.32019460743274397174e+00,
 269         6.52081912033011246166e+00, 6.72781213889469142941e+00,
 270         6.94137582119703555605e+00, 7.16171874249371143151e+00,
 271         1.00000000000000000000e+00, 7.38905609893065040694e+00,
 272         5.45981500331442362040e+01, 4.03428793492735110249e+02,
 273         2.98095798704172830185e+03, 2.20264657948067178950e+04,
 274         1.62754791419003915507e+05, 1.20260428416477679275e+06,
 275         8.88611052050787210464e+06, 6.56599691373305097222e+07,
 276         4.85165195409790277481e+08, 3.58491284613159179688e+09,
 277         2.64891221298434715271e+10, 1.95729609428838775635e+11,
 278         1.44625706429147509766e+12, 1.06864745815244628906e+13,
 279         7.89629601826806875000e+13, 5.83461742527454875000e+14,
 280         4.31123154711519500000e+15, 3.18559317571137560000e+16,
 281         2.35385266837020000000e+17, 1.73927494152050099200e+18,
 282         1.28516001143593082880e+19, 9.49611942060244828160e+19,
 283         7.01673591209763143680e+20, 5.18470552858707204506e+21,
 284         3.83100800071657691546e+22, 2.83075330327469394756e+23,
 285         2.09165949601299610311e+24, 1.54553893559010391826e+25,
 286         1.14200738981568423454e+26, 8.43835666874145383188e+26,
 287         6.23514908081161674391e+27, 4.60718663433129178064e+28,
 288         3.40427604993174075827e+29, 2.51543867091916687979e+30,
 289         1.85867174528412788702e+31, 1.37338297954017610775e+32,
 290         1.01480038811388874615e+33, 7.49841699699012090701e+33,
 291         5.54062238439350983445e+34, 4.09399696212745451138e+35,
 292         3.02507732220114256223e+36, 2.23524660373471497416e+37,
 293         1.65163625499400180987e+38, 1.22040329431784083418e+39,
 294         9.01762840503429851945e+39, 6.66317621641089618500e+40,
 295         4.92345828601205826106e+41, 3.63797094760880474988e+42,
 296         2.68811714181613560943e+43, 1.98626483613765434356e+44,
 297         1.46766223015544238535e+45, 1.08446385529002313207e+46,
 298         8.01316426400059069850e+46, 5.92097202766466993617e+47,
 299         4.37503944726134096988e+48, 3.23274119108485947460e+49,
 300         2.38869060142499127023e+50, 1.76501688569176554670e+51,
 301         1.30418087839363225614e+52, 9.63666567360320166416e+52,
 302         7.12058632688933793173e+53, 5.26144118266638596909e+54,
 303 };
 304
 305 /*
 306  * EN[k] = exp(-2^(k-5))
 307  */
 308 static const double EN[] = {
 309         9.69233234476344129860e-01, 9.39413062813475807644e-01,
 310         8.82496902584595455110e-01, 7.78800783071404878477e-01,
 311         6.06530659712633424263e-01, 3.67879441171442334024e-01,
 312         1.35335283236612702318e-01, 1.83156388887341786686e-02,
 313         3.35462627902511853224e-04, 1.12535174719259116458e-07,
 314         1.26641655490941755372e-14, 1.60381089054863792659e-28,
 315 #if defined(FDTOS_TRAPS_INCOMPLETE_IN_FNS_MODE)
 316         2.96555550007072683578e-38,     /* exp(-128) scaled up by 2^60 */
 317 #else
 318         2.57220937264241481170e-56,
 319 #endif
 320 };
 321
 322 static const float F[] = {
 323         0.0f,
 324         1.0f,
 325         5.0000000951292138e-01F,
 326         1.6666518897347284e-01F,
 327         3.4028234663852885981170E+38F,
 328         1.1754943508222875079688E-38F,
 329 #if defined(FDTOS_TRAPS_INCOMPLETE_IN_FNS_MODE)
 330         8.67361737988403547205962240695953369140625e-19F
 331 #endif
 332 };
 333
 334 #define zero    F[0]
 335 #define one     F[1]
 336 #define p1      F[2]
 337 #define p2      F[3]
 338 #define big     F[4]
 339 #define tiny    F[5]
 340 #if defined(FDTOS_TRAPS_INCOMPLETE_IN_FNS_MODE)
 341 #define twom60  F[6]
 342 #endif
 343
 344 float
 345 expf(float xf) {
 346         double  w, p, q;
 347         int     hx, ix, n;
 348
 349         hx = *(int *)&xf;
 350         ix = hx & ~0x80000000;
 351
 352         if (ix < 0x3c800000) {  /* |x| < 2**-6 */
 353                 if (ix < 0x38800000)    /* |x| < 2**-14 */
 354                         return (one + xf);
 355                 return (one + (xf + (xf * xf) * (p1 + xf * p2)));
 356         }
 357
 358         n = ix >> 23;           /* biased exponent */
 359
 360         if (n >= 0x86) {        /* |x| >= 2^7 */
 361                 if (n >= 0xff) {        /* x is nan of +-inf */
 362                         if (hx == 0xff800000)
 363                                 return (zero);  /* exp(-inf)=0 */
 364                         return (xf * xf);       /* exp(nan/inf) is nan or inf */
 365                 }
 366                 if (hx > 0)
 367                         return (big * big);     /* overflow */
 368                 else
 369                         return (tiny * tiny);   /* underflow */
 370         }
 371
 372         ix -= n << 23;
 373         if (hx > 0)
 374                 ix += 0x800000;
 375         else
 376                 ix = 0x800000 - ix;
 377         if (n >= 0x7f) {        /* n >= 0 */
 378                 ix <<= n - 0x7f;
 379                 w = ET[(ix & 0x3f) + 64] * ET[((ix >> 6) & 0x3f) + 128];
 380                 p = ET[((ix >> 12) & 0x3f) + 192] *
 381                     ET[((ix >> 18) & 0x3f) + 256];
 382                 q = ET[((ix >> 24) & 0x3f) + 320];
 383         } else {
 384                 ix <<= n - 0x79;
 385                 w = ET[ix & 0x3f] * ET[((ix >> 6) & 0x3f) + 64];
 386                 p = ET[((ix >> 12) & 0x3f) + 128] *
 387                     ET[((ix >> 18) & 0x3f) + 192];
 388                 q = ET[((ix >> 24) & 0x3f) + 256];
 389         }
 390         xf = (float)((w * p) * (hx < 0 ? q * EN[n - 0x79] : q));
 391 #if defined(FDTOS_TRAPS_INCOMPLETE_IN_FNS_MODE)
 392         if ((unsigned)hx >= 0xc2800000u) {
 393                 if ((unsigned)hx >= 0xc2aeac50) { /* force underflow */
 394                         volatile float  t = tiny;
 395                         t *= t;
 396                 }
 397                 return (xf * twom60);
 398         }
 399 #endif
 400         return (xf);
 401 }