expf.c revision 25c28e83beb90e7c80452a7c818c5e6f73a07dc8
/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
/* INDENT OFF */
/*
* float expf(float x);
* Code by K.C. Ng for SUN 5.0 libmopt
* 11/5/99
* Method :
* More precisely:
* x > 88.722839355...(0x42B17218) => overflow;
* x < -103.97207642..(0xc2CFF1B4) => underflow.
* 2. For |x| < 2^-6, use polynomail
* exp(x) = 1 + x + p1*x^2 + p2*x^3
* 3. Otherwise, write |x|=(1+r)*2^n, where 0<=r<1.
* Let t = 2^n * (1+r) .... x > 0;
* t = 2^n * (1-r) .... x < 0. (x= -2**(n+1)+t)
* Since -6 <= n <= 6, we may break t into
* six 6-bits chunks:
* -5 -11 -17 -23 -29
* t=j *2+j *2 +j *2 +j *2 +j *2 +j *2
* 1 2 3 4 5 6
*
* where 0 <= j < 64 for i = 1,...,6.
* i
* Note that since t has only 24 significant bits,
* either j or j must be 0.
* 1 6
* 7-6i
* One may define j by (int) ( t * 2 ) mod 64
* i
* mathematically. In actual implementation, they can
* be obtained by manipulating the exponent and
* mantissa bits as follow:
* Let ix = (HEX(x)&0x007fffff)|0x00800000.
* If n>=0, let ix=ix<<n, then j =0 and
* 6
* j = ix>>(30-6i)) mod 64 ...i=1,...,5
* i
* Otherwise, let ix=ix<<(j+6), then j = 0 and
* 1
* j = ix>>(36-6i)) mod 64 ...i=2,...,6
* i
*
* 4. Compute exp(t) by table look-up method.
* Precompute ET[k] = exp(j*2^(7-6i)), k=j+64*(6-i).
* Then
* exp(t) = ET[j +320]*ET[j +256]*ET[j +192]*
* 1 2 3
*
* ET[j +128]*ET[j +64]*ET[j ]
* 4 5 6
*
* n+1
* 5. If x < 0, return exp(-2 )* exp(t). Note that
* -6 <= n <= 6. Let k = n - 6, then we can
* precompute
* k-5 n+1
* EN[k] = exp(-2 ) = exp(-2 ) for k=0,1,...,12.
*
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF) = 0;
* for finite argument, only exp(0) = 1 is exact.
*
* Accuracy:
* All calculations are done in double precision except for
* the case |x| < 2^-6. When |x| < 2^-6, the error is less
* than 0.55 ulp. When |x| >= 2^-6 and the result is normal,
* the error is less than 0.51 ulp. When FDTOS_TRAPS_... is
* defined and the result is subnormal, the error can be as
* large as 0.75 ulp.
*/
/* INDENT ON */
#include "libm.h"
/*
* ET[k] = exp(j*2^(7-6i)) , where j = k mod 64, i = k/64
*/
static const double ET[] = {
1.00000000000000000000e+00, 1.00000000186264514923e+00,
1.00000000372529029846e+00, 1.00000000558793544769e+00,
1.00000000745058059692e+00, 1.00000000931322574615e+00,
1.00000001117587089539e+00, 1.00000001303851604462e+00,
1.00000001490116119385e+00, 1.00000001676380656512e+00,
1.00000001862645171435e+00, 1.00000002048909686359e+00,
1.00000002235174201282e+00, 1.00000002421438716205e+00,
1.00000002607703253332e+00, 1.00000002793967768255e+00,
1.00000002980232283178e+00, 1.00000003166496798102e+00,
1.00000003352761335229e+00, 1.00000003539025850152e+00,
1.00000003725290365075e+00, 1.00000003911554879998e+00,
1.00000004097819417126e+00, 1.00000004284083932049e+00,
1.00000004470348446972e+00, 1.00000004656612984100e+00,
1.00000004842877499023e+00, 1.00000005029142036150e+00,
1.00000005215406551073e+00, 1.00000005401671088201e+00,
1.00000005587935603124e+00, 1.00000005774200140252e+00,
1.00000005960464655175e+00, 1.00000006146729192302e+00,
1.00000006332993707225e+00, 1.00000006519258244353e+00,
1.00000006705522759276e+00, 1.00000006891787296404e+00,
1.00000007078051811327e+00, 1.00000007264316348454e+00,
1.00000007450580863377e+00, 1.00000007636845400505e+00,
1.00000007823109937632e+00, 1.00000008009374452556e+00,
1.00000008195638989683e+00, 1.00000008381903526811e+00,
1.00000008568168063938e+00, 1.00000008754432578861e+00,
1.00000008940697115989e+00, 1.00000009126961653116e+00,
1.00000009313226190244e+00, 1.00000009499490705167e+00,
1.00000009685755242295e+00, 1.00000009872019779422e+00,
1.00000010058284316550e+00, 1.00000010244548853677e+00,
1.00000010430813368600e+00, 1.00000010617077905728e+00,
1.00000010803342442856e+00, 1.00000010989606979983e+00,
1.00000011175871517111e+00, 1.00000011362136054238e+00,
1.00000011548400591366e+00, 1.00000011734665128493e+00,
1.00000000000000000000e+00, 1.00000011920929665621e+00,
1.00000023841860752327e+00, 1.00000035762793260119e+00,
1.00000047683727188996e+00, 1.00000059604662538959e+00,
1.00000071525599310007e+00, 1.00000083446537502141e+00,
1.00000095367477115360e+00, 1.00000107288418149665e+00,
1.00000119209360605055e+00, 1.00000131130304481530e+00,
1.00000143051249779091e+00, 1.00000154972196497738e+00,
1.00000166893144637470e+00, 1.00000178814094198287e+00,
1.00000190735045180190e+00, 1.00000202655997583179e+00,
1.00000214576951407253e+00, 1.00000226497906652412e+00,
1.00000238418863318657e+00, 1.00000250339821405987e+00,
1.00000262260780914403e+00, 1.00000274181741843904e+00,
1.00000286102704194491e+00, 1.00000298023667966163e+00,
1.00000309944633158921e+00, 1.00000321865599772764e+00,
1.00000333786567807692e+00, 1.00000345707537263706e+00,
1.00000357628508140806e+00, 1.00000369549480438991e+00,
1.00000381470454158261e+00, 1.00000393391429298617e+00,
1.00000405312405860059e+00, 1.00000417233383842586e+00,
1.00000429154363246198e+00, 1.00000441075344070896e+00,
1.00000452996326316679e+00, 1.00000464917309983548e+00,
1.00000476838295071502e+00, 1.00000488759281580542e+00,
1.00000500680269510667e+00, 1.00000512601258861878e+00,
1.00000524522249634174e+00, 1.00000536443241827556e+00,
1.00000548364235442023e+00, 1.00000560285230477575e+00,
1.00000572206226934213e+00, 1.00000584127224811937e+00,
1.00000596048224110746e+00, 1.00000607969224830640e+00,
1.00000619890226971620e+00, 1.00000631811230533685e+00,
1.00000643732235516836e+00, 1.00000655653241921073e+00,
1.00000667574249746394e+00, 1.00000679495258992802e+00,
1.00000691416269660294e+00, 1.00000703337281748873e+00,
1.00000715258295258536e+00, 1.00000727179310189285e+00,
1.00000739100326541120e+00, 1.00000751021344314040e+00,
1.00000000000000000000e+00, 1.00000762942363508046e+00,
1.00001525890547848796e+00, 1.00002288844553022251e+00,
1.00003051804379095024e+00, 1.00003814770026133729e+00,
1.00004577741494138365e+00, 1.00005340718783175546e+00,
1.00006103701893311886e+00, 1.00006866690824547383e+00,
1.00007629685576948653e+00, 1.00008392686150582307e+00,
1.00009155692545448346e+00, 1.00009918704761613384e+00,
1.00010681722799144033e+00, 1.00011444746658040295e+00,
1.00012207776338368781e+00, 1.00012970811840196106e+00,
1.00013733853163522269e+00, 1.00014496900308413885e+00,
1.00015259953274937565e+00, 1.00016023012063093311e+00,
1.00016786076672947736e+00, 1.00017549147104567453e+00,
1.00018312223357952462e+00, 1.00019075305433191581e+00,
1.00019838393330284809e+00, 1.00020601487049298761e+00,
1.00021364586590300050e+00, 1.00022127691953288675e+00,
1.00022890803138353455e+00, 1.00023653920145494389e+00,
1.00024417042974778091e+00, 1.00025180171626271175e+00,
1.00025943306099973640e+00, 1.00026706446395974304e+00,
1.00027469592514273167e+00, 1.00028232744454959047e+00,
1.00028995902218031944e+00, 1.00029759065803558471e+00,
1.00030522235211605242e+00, 1.00031285410442172257e+00,
1.00032048591495348333e+00, 1.00032811778371155675e+00,
1.00033574971069616488e+00, 1.00034338169590819589e+00,
1.00035101373934764979e+00, 1.00035864584101541475e+00,
1.00036627800091149076e+00, 1.00037391021903676602e+00,
1.00038154249539146257e+00, 1.00038917482997580244e+00,
1.00039680722279067382e+00, 1.00040443967383629875e+00,
1.00041207218311289928e+00, 1.00041970475062136359e+00,
1.00042733737636191371e+00, 1.00043497006033499375e+00,
1.00044260280254104778e+00, 1.00045023560298029786e+00,
1.00045786846165363215e+00, 1.00046550137856127272e+00,
1.00047313435370366363e+00, 1.00048076738708124900e+00,
1.00000000000000000000e+00, 1.00048840047869447289e+00,
1.00097703949241645383e+00, 1.00146591715766675179e+00,
1.00195503359100279717e+00, 1.00244438890903908579e+00,
1.00293398322844673487e+00, 1.00342381666595459322e+00,
1.00391388933834746489e+00, 1.00440420136246855165e+00,
1.00489475285521656645e+00, 1.00538554393354861993e+00,
1.00587657471447822211e+00, 1.00636784531507639251e+00,
1.00685935585247099411e+00, 1.00735110644384739942e+00,
1.00784309720644804642e+00, 1.00833532825757243856e+00,
1.00882779971457803292e+00, 1.00932051169487890796e+00,
1.00981346431594687374e+00, 1.01030665769531102782e+00,
1.01080009195055753324e+00, 1.01129376719933050666e+00,
1.01178768355933157430e+00, 1.01228184114831898377e+00,
1.01277624008410960244e+00, 1.01327088048457714109e+00,
1.01376576246765282008e+00, 1.01426088615132625748e+00,
1.01475625165364347069e+00, 1.01525185909270931894e+00,
1.01574770858668572693e+00, 1.01624380025379235093e+00,
1.01674013421230657883e+00, 1.01723671058056375216e+00,
1.01773352947695694404e+00, 1.01823059101993673714e+00,
1.01872789532801233392e+00, 1.01922544251975000229e+00,
1.01972323271377418585e+00, 1.02022126602876750390e+00,
1.02071954258347008526e+00, 1.02121806249668067856e+00,
1.02171682588725554197e+00, 1.02221583287410910934e+00,
1.02271508357621376817e+00, 1.02321457811260052573e+00,
1.02371431660235789884e+00, 1.02421429916463280207e+00,
1.02471452591863054771e+00, 1.02521499698361440167e+00,
1.02571571247890602763e+00, 1.02621667252388526492e+00,
1.02671787723799012859e+00, 1.02721932674071725344e+00,
1.02772102115162167202e+00, 1.02822296059031659254e+00,
1.02872514517647339893e+00, 1.02922757502982276101e+00,
1.02973025027015285815e+00, 1.03023317101731093359e+00,
1.03073633739120262831e+00, 1.03123974951179242510e+00,
1.00000000000000000000e+00, 1.03174340749910276038e+00,
1.06449445891785954288e+00, 1.09828514030782575794e+00,
1.13314845306682632220e+00, 1.16911844616950433284e+00,
1.20623024942098067136e+00, 1.24452010776609522935e+00,
1.28402541668774139438e+00, 1.32478475872886569675e+00,
1.36683794117379631139e+00, 1.41022603492571074746e+00,
1.45499141461820125087e+00, 1.50117780000012279729e+00,
1.54883029863413312910e+00, 1.59799544995063325104e+00,
1.64872127070012819416e+00, 1.70105730184840076014e+00,
1.75505465696029849809e+00, 1.81076607211938722664e+00,
1.86824595743222232613e+00, 1.92755045016754467113e+00,
1.98873746958229191684e+00, 2.05186677348797674725e+00,
2.11700001661267478426e+00, 2.18420081081561789915e+00,
2.25353478721320854561e+00, 2.32506966027712103084e+00,
2.39887529396709808793e+00, 2.47502376996302508871e+00,
2.55358945806292680913e+00, 2.63464908881563086851e+00,
2.71828182845904553488e+00, 2.80456935623722669604e+00,
2.89359594417176113623e+00, 2.98544853936535581340e+00,
3.08021684891803104733e+00, 3.17799342753883840018e+00,
3.27887376793867346692e+00, 3.38295639409246895468e+00,
3.49034295746184142217e+00, 3.60113833627217561073e+00,
3.71545073794110392029e+00, 3.83339180475841034834e+00,
3.95507672292057721464e+00, 4.08062433502646015882e+00,
4.21015725614395996956e+00, 4.34380199356104235164e+00,
4.48168907033806451778e+00, 4.62395315278208052234e+00,
4.77073318196760265408e+00, 4.92217250943229078786e+00,
5.07841903718008147450e+00, 5.23962536212848917216e+00,
5.40594892514116676097e+00, 5.57755216479125959239e+00,
5.75460267600573072144e+00, 5.93727337374560715233e+00,
6.12574266188198635064e+00, 6.32019460743274397174e+00,
6.52081912033011246166e+00, 6.72781213889469142941e+00,
6.94137582119703555605e+00, 7.16171874249371143151e+00,
1.00000000000000000000e+00, 7.38905609893065040694e+00,
5.45981500331442362040e+01, 4.03428793492735110249e+02,
2.98095798704172830185e+03, 2.20264657948067178950e+04,
1.62754791419003915507e+05, 1.20260428416477679275e+06,
8.88611052050787210464e+06, 6.56599691373305097222e+07,
4.85165195409790277481e+08, 3.58491284613159179688e+09,
2.64891221298434715271e+10, 1.95729609428838775635e+11,
1.44625706429147509766e+12, 1.06864745815244628906e+13,
7.89629601826806875000e+13, 5.83461742527454875000e+14,
4.31123154711519500000e+15, 3.18559317571137560000e+16,
2.35385266837020000000e+17, 1.73927494152050099200e+18,
1.28516001143593082880e+19, 9.49611942060244828160e+19,
7.01673591209763143680e+20, 5.18470552858707204506e+21,
3.83100800071657691546e+22, 2.83075330327469394756e+23,
2.09165949601299610311e+24, 1.54553893559010391826e+25,
1.14200738981568423454e+26, 8.43835666874145383188e+26,
6.23514908081161674391e+27, 4.60718663433129178064e+28,
3.40427604993174075827e+29, 2.51543867091916687979e+30,
1.85867174528412788702e+31, 1.37338297954017610775e+32,
1.01480038811388874615e+33, 7.49841699699012090701e+33,
5.54062238439350983445e+34, 4.09399696212745451138e+35,
3.02507732220114256223e+36, 2.23524660373471497416e+37,
1.65163625499400180987e+38, 1.22040329431784083418e+39,
9.01762840503429851945e+39, 6.66317621641089618500e+40,
4.92345828601205826106e+41, 3.63797094760880474988e+42,
2.68811714181613560943e+43, 1.98626483613765434356e+44,
1.46766223015544238535e+45, 1.08446385529002313207e+46,
8.01316426400059069850e+46, 5.92097202766466993617e+47,
4.37503944726134096988e+48, 3.23274119108485947460e+49,
2.38869060142499127023e+50, 1.76501688569176554670e+51,
1.30418087839363225614e+52, 9.63666567360320166416e+52,
7.12058632688933793173e+53, 5.26144118266638596909e+54,
};
/*
* EN[k] = exp(-2^(k-5))
*/
static const double EN[] = {
9.69233234476344129860e-01, 9.39413062813475807644e-01,
8.82496902584595455110e-01, 7.78800783071404878477e-01,
6.06530659712633424263e-01, 3.67879441171442334024e-01,
1.35335283236612702318e-01, 1.83156388887341786686e-02,
3.35462627902511853224e-04, 1.12535174719259116458e-07,
1.26641655490941755372e-14, 1.60381089054863792659e-28,
#if defined(FDTOS_TRAPS_INCOMPLETE_IN_FNS_MODE)
2.96555550007072683578e-38, /* exp(-128) scaled up by 2^60 */
#else
2.57220937264241481170e-56,
#endif
};
static const float F[] = {
0.0f,
1.0f,
5.0000000951292138e-01F,
1.6666518897347284e-01F,
3.4028234663852885981170E+38F,
1.1754943508222875079688E-38F,
#if defined(FDTOS_TRAPS_INCOMPLETE_IN_FNS_MODE)
8.67361737988403547205962240695953369140625e-19F
#endif
};
#define zero F[0]
#define one F[1]
#define p1 F[2]
#define p2 F[3]
#define big F[4]
#define tiny F[5]
#if defined(FDTOS_TRAPS_INCOMPLETE_IN_FNS_MODE)
#define twom60 F[6]
#endif
float
double w, p, q;
}
if (n >= 0x86) { /* |x| >= 2^7 */
if (n >= 0xff) { /* x is nan of +-inf */
if (hx == 0xff800000)
return (zero); /* exp(-inf)=0 */
}
if (hx > 0)
else
}
ix -= n << 23;
if (hx > 0)
ix += 0x800000;
else
if (n >= 0x7f) { /* n >= 0 */
ix <<= n - 0x7f;
} else {
ix <<= n - 0x79;
}
#if defined(FDTOS_TRAPS_INCOMPLETE_IN_FNS_MODE)
if ((unsigned)hx >= 0xc2800000u) {
volatile float t = tiny;
t *= t;
}
}
#endif
return (xf);
}