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 }