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 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __tgammaf = tgammaf
32
33 /*
34 * True gamma function
35 *
36 * float tgammaf(float x)
37 *
38 * Algorithm: see tgamma.c
39 *
40 * Maximum error observed: 0.87ulp (both positive and negative arguments)
41 */
42
43 #include "libm.h"
44 #include <math.h>
45 #if defined(__SUNPRO_C)
46 #include <sunmath.h>
47 #endif
48 #include <sys/isa_defs.h>
49
50 #if defined(_BIG_ENDIAN)
51 #define HIWORD 0
52 #define LOWORD 1
53 #else
54 #define HIWORD 1
55 #define LOWORD 0
56 #endif
57 #define __HI(x) ((int *)&x)[HIWORD]
58 #define __LO(x) ((unsigned *)&x)[LOWORD]
59
60 /* Coefficients for primary intervals GTi() */
61 static const double cr[] = {
62 /* p1 */
63 +7.09087253435088360271451613398019280077561279443e-0001,
64 -5.17229560788652108545141978238701790105241761089e-0001,
65 +5.23403394528150789405825222323770647162337764327e-0001,
66 -4.54586308717075010784041566069480411732634814899e-0001,
67 +4.20596490915239085459964590559256913498190955233e-0001,
68 -3.57307589712377520978332185838241458642142185789e-0001,
69 /* p2 */
70 +4.28486983980295198166056119223984284434264344578e-0001,
71 -1.30704539487709138528680121627899735386650103914e-0001,
72 +1.60856285038051955072861219352655851542955430871e-0001,
73 -9.22285161346010583774458802067371182158937943507e-0002,
74 +7.19240511767225260740890292605070595560626179357e-0002,
75 -4.88158265593355093703112238534484636193260459574e-0002,
76 /* p3 */
77 +3.82409531118807759081121479786092134814808872880e-0001,
78 +2.65309888180188647956400403013495759365167853426e-0002,
79 +8.06815109775079171923561169415370309376296739835e-0002,
80 -1.54821591666137613928840890835174351674007764799e-0002,
81 +1.76308239242717268530498313416899188157165183405e-0002,
82 /* GZi and TZi */
83 +0.9382046279096824494097535615803269576988, /* GZ1 */
84 +0.8856031944108887002788159005825887332080, /* GZ2 */
85 +0.9367814114636523216188468970808378497426, /* GZ3 */
86 -0.3517214357852935791015625, /* TZ1 */
87 +0.280530631542205810546875, /* TZ3 */
88 };
89
90 #define P10 cr[0]
91 #define P11 cr[1]
92 #define P12 cr[2]
93 #define P13 cr[3]
94 #define P14 cr[4]
95 #define P15 cr[5]
96 #define P20 cr[6]
97 #define P21 cr[7]
98 #define P22 cr[8]
99 #define P23 cr[9]
100 #define P24 cr[10]
101 #define P25 cr[11]
102 #define P30 cr[12]
103 #define P31 cr[13]
104 #define P32 cr[14]
105 #define P33 cr[15]
106 #define P34 cr[16]
107 #define GZ1 cr[17]
108 #define GZ2 cr[18]
109 #define GZ3 cr[19]
110 #define TZ1 cr[20]
111 #define TZ3 cr[21]
112
113 /* compute gamma(y) for y in GT1 = [1.0000, 1.2845] */
114 static double
115 GT1(double y)
116 {
117 double z, r;
118
119 z = y * y;
120 r = TZ1 * y + z * ((P10 + y * P11 + z * P12) + (z * y) * (P13 + y *
121 P14 + z * P15));
122 return (GZ1 + r);
123 }
124
125 /* compute gamma(y) for y in GT2 = [1.2844, 1.6374] */
126 static double
127 GT2(double y)
128 {
129 double z;
130
131 z = y * y;
132 return (GZ2 + z * ((P20 + y * P21 + z * P22) + (z * y) * (P23 + y *
133 P24 + z * P25)));
134 }
135
136 /* compute gamma(y) for y in GT3 = [1.6373, 2.0000] */
137 static double
138 GT3(double y)
139 {
140 double z, r;
141
142 z = y * y;
143 r = TZ3 * y + z * ((P30 + y * P31 + z * P32) + (z * y) * (P33 + y *
144 P34));
145 return (GZ3 + r);
146 }
147
148 static const double c[] = {
149 +1.0,
150 +2.0,
151 +0.5,
152 +1.0e-300,
153 +6.666717231848518054693623697539230e-0001, /* A1=T3[0] */
154 +8.33333330959694065245736888749042811909994573178e-0002, /* GP[0] */
155 -2.77765545601667179767706600890361535225507762168e-0003, /* GP[1] */
156 +7.77830853479775281781085278324621033523037489883e-0004, /* GP[2] */
157 +4.18938533204672741744150788368695779923320328369e-0001, /* hln2pi */
158 +2.16608493924982901946e-02, /* ln2_32 */
159 +4.61662413084468283841e+01, /* invln2_32 */
160 +5.00004103388988968841156421415669985414073453720e-0001, /* Et1 */
161 +1.66667656752800761782778277828110208108687545908e-0001, /* Et2 */
162 };
163
164 #define one c[0]
165 #define two c[1]
166 #define half c[2]
167 #define tiny c[3]
168 #define A1 c[4]
169 #define GP0 c[5]
170 #define GP1 c[6]
171 #define GP2 c[7]
172 #define hln2pi c[8]
173 #define ln2_32 c[9]
174 #define invln2_32 c[10]
175 #define Et1 c[11]
176 #define Et2 c[12]
177
178 /* S[j] = 2**(j/32.) for the final computation of exp(w) */
179 static const double S[] = {
180 +1.00000000000000000000e+00, /* 3FF0000000000000 */
181 +1.02189714865411662714e+00, /* 3FF059B0D3158574 */
182 +1.04427378242741375480e+00, /* 3FF0B5586CF9890F */
183 +1.06714040067682369717e+00, /* 3FF11301D0125B51 */
184 +1.09050773266525768967e+00, /* 3FF172B83C7D517B */
185 +1.11438674259589243221e+00, /* 3FF1D4873168B9AA */
186 +1.13878863475669156458e+00, /* 3FF2387A6E756238 */
187 +1.16372485877757747552e+00, /* 3FF29E9DF51FDEE1 */
188 +1.18920711500272102690e+00, /* 3FF306FE0A31B715 */
189 +1.21524735998046895524e+00, /* 3FF371A7373AA9CB */
190 +1.24185781207348400201e+00, /* 3FF3DEA64C123422 */
191 +1.26905095719173321989e+00, /* 3FF44E086061892D */
192 +1.29683955465100964055e+00, /* 3FF4BFDAD5362A27 */
193 +1.32523664315974132322e+00, /* 3FF5342B569D4F82 */
194 +1.35425554693689265129e+00, /* 3FF5AB07DD485429 */
195 +1.38390988196383202258e+00, /* 3FF6247EB03A5585 */
196 +1.41421356237309514547e+00, /* 3FF6A09E667F3BCD */
197 +1.44518080697704665027e+00, /* 3FF71F75E8EC5F74 */
198 +1.47682614593949934623e+00, /* 3FF7A11473EB0187 */
199 +1.50916442759342284141e+00, /* 3FF82589994CCE13 */
200 +1.54221082540794074411e+00, /* 3FF8ACE5422AA0DB */
201 +1.57598084510788649659e+00, /* 3FF93737B0CDC5E5 */
202 +1.61049033194925428347e+00, /* 3FF9C49182A3F090 */
203 +1.64575547815396494578e+00, /* 3FFA5503B23E255D */
204 +1.68179283050742900407e+00, /* 3FFAE89F995AD3AD */
205 +1.71861929812247793414e+00, /* 3FFB7F76F2FB5E47 */
206 +1.75625216037329945351e+00, /* 3FFC199BDD85529C */
207 +1.79470907500310716820e+00, /* 3FFCB720DCEF9069 */
208 +1.83400808640934243066e+00, /* 3FFD5818DCFBA487 */
209 +1.87416763411029996256e+00, /* 3FFDFC97337B9B5F */
210 +1.91520656139714740007e+00, /* 3FFEA4AFA2A490DA */
211 +1.95714412417540017941e+00, /* 3FFF50765B6E4540 */
212 };
213
214
215 /*
216 * return tgammaf(x) in double for 8<x<=35.040096283... using Stirling's formula
217 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
218 */
219
220 /*
221 * compute ss = log(x)-1
222 *
223 * log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
224 * j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
225 * T1(n-3) = n*log(2)-1, n=3,4,5
226 * T2(j) = log(z[j]),
227 * T3(s) = 2s + A1*s^3
228 * Note
229 * (1) Remez error for T3(s) is bounded by 2**(-35.8)
230 * (see mpremez/work/Log/tgamma_log_2_outr1)
231 */
232 static const double T1[] = { /* T1[j]=(j+3)*log(2)-1 */
233 +1.079441541679835928251696364375e+00,
234 +1.772588722239781237668928485833e+00,
235 +2.465735902799726547086160607291e+00,
236 };
237
238 static const double T2[] = { /* T2[j]=log(1+j/64+1/128) */
239 +7.782140442054948947462900061137e-03,
240 +2.316705928153437822879916096229e-02,
241 +3.831886430213659919375532512380e-02,
242 +5.324451451881228286587019378653e-02,
243 +6.795066190850774939456527777263e-02,
244 +8.244366921107459126816006866831e-02,
245 +9.672962645855111229557105648746e-02,
246 +1.108143663402901141948061693232e-01,
247 +1.247034785009572358634065153809e-01,
248 +1.384023228591191356853258736016e-01,
249 +1.519160420258419750718034248969e-01,
250 +1.652495728953071628756114492772e-01,
251 +1.784076574728182971194002415109e-01,
252 +1.913948529996294546092988075613e-01,
253 +2.042155414286908915038203861962e-01,
254 +2.168739383006143596190895257443e-01,
255 +2.293741010648458299914807250461e-01,
256 +2.417199368871451681443075159135e-01,
257 +2.539152099809634441373232979066e-01,
258 +2.659635484971379413391259265375e-01,
259 +2.778684510034563061863500329234e-01,
260 +2.896332925830426768788930555257e-01,
261 +3.012613305781617810128755382338e-01,
262 +3.127557100038968883862465596883e-01,
263 +3.241194686542119760906707604350e-01,
264 +3.353555419211378302571795798142e-01,
265 +3.464667673462085809184621884258e-01,
266 +3.574558889218037742260094901409e-01,
267 +3.683255611587076530482301540504e-01,
268 +3.790783529349694583908533456310e-01,
269 +3.897167511400252133704636040035e-01,
270 +4.002431641270127069293251019951e-01,
271 +4.106599249852683859343062031758e-01,
272 +4.209692946441296361288671615068e-01,
273 +4.311734648183713408591724789556e-01,
274 +4.412745608048752294894964416613e-01,
275 +4.512746441394585851446923830790e-01,
276 +4.611757151221701663679999255979e-01,
277 +4.709797152187910125468978560564e-01,
278 +4.806885293457519076766184554480e-01,
279 +4.903039880451938381503461596457e-01,
280 +4.998278695564493298213314152470e-01,
281 +5.092619017898079468040749192283e-01,
282 +5.186077642080456321529769963648e-01,
283 +5.278670896208423851138922177783e-01,
284 +5.370414658968836545667292441538e-01,
285 +5.461324375981356503823972092312e-01,
286 +5.551415075405015927154803595159e-01,
287 +5.640701382848029660713842900902e-01,
288 +5.729197535617855090927567266263e-01,
289 +5.816917396346224825206107537254e-01,
290 +5.903874466021763746419167081236e-01,
291 +5.990081896460833993816000244617e-01,
292 +6.075552502245417955010851527911e-01,
293 +6.160298772155140196475659281967e-01,
294 +6.244332880118935010425387440547e-01,
295 +6.327666695710378295457864685036e-01,
296 +6.410311794209312910556013344054e-01,
297 +6.492279466251098188908399699053e-01,
298 +6.573580727083600301418900232459e-01,
299 +6.654226325450904489500926100067e-01,
300 +6.734226752121667202979603888010e-01,
301 +6.813592248079030689480715595681e-01,
302 +6.892332812388089803249143378146e-01,
303 };
304
305 static double
306 large_gam(double x)
307 {
308 double ss, zz, z, t1, t2, w, y, u;
309 unsigned lx;
310 int k, ix, j, m;
311
312 ix = __HI(x);
313 lx = __LO(x);
314 m = (ix >> 20) - 0x3ff; /* exponent of x, range:3-5 */
315 ix = (ix & 0x000fffff) | 0x3ff00000; /* y = scale x to [1,2] */
316 __HI(y) = ix;
317 __LO(y) = lx;
318 __HI(z) = (ix & 0xffffc000) | 0x2000; /* z[j]=1+j/64+1/128 */
319 __LO(z) = 0;
320 j = (ix >> 14) & 0x3f;
321 t1 = y + z;
322 t2 = y - z;
323 u = t2 / t1;
324 ss = T1[m - 3] + T2[j] + u * (two + A1 * (u * u));
325 /* ss = log(x)-1 */
326
327 /*
328 * compute ww = (x-.5)*(log(x)-1) + .5*(log(2pi)-1) + 1/x*(P(1/x^2)))
329 * where ss = log(x) - 1
330 */
331 z = one / x;
332 zz = z * z;
333 w = ((x - half) * ss + hln2pi) + z * (GP0 + zz * GP1 + (zz * zz) * GP2);
334 k = (int)(w * invln2_32 + half);
335
336 /* compute the exponential of w */
337 j = k & 0x1f;
338 m = k >> 5;
339 z = w - (double)k * ln2_32;
340 zz = S[j] * (one + z + (z * z) * (Et1 + z * Et2));
341 __HI(zz) += m << 20;
342 return (zz);
343 }
344
345
346 /*
347 * kpsin(x)= sin(pi*x)/pi
348 * 3 5 7 9
349 * = x+ks[0]*x +ks[1]*x +ks[2]*x +ks[3]*x
350 */
351 static const double ks[] = {
352 -1.64493404985645811354476665052005342839447790544e+0000,
353 +8.11740794458351064092797249069438269367389272270e-0001,
354 -1.90703144603551216933075809162889536878854055202e-0001,
355 +2.55742333994264563281155312271481108635575331201e-0002,
356 };
357
358 static double
359 kpsin(double x)
360 {
361 double z;
362
363 z = x * x;
364 return (x + (x * z) * ((ks[0] + z * ks[1]) + (z * z) * (ks[2] + z *
365 ks[3])));
366 }
367
368
369 /*
370 * kpcos(x)= cos(pi*x)/pi
371 * 2 4 6
372 * = kc[0]+kc[1]*x +kc[2]*x +kc[3]*x
373 */
374 static const double kc[] = {
375 +3.18309886183790671537767526745028724068919291480e-0001,
376 -1.57079581447762568199467875065854538626594937791e+0000,
377 +1.29183528092558692844073004029568674027807393862e+0000,
378 -4.20232949771307685981015914425195471602739075537e-0001,
379 };
380
381 static double
382 kpcos(double x)
383 {
384 double z;
385
386 z = x * x;
387 return (kc[0] + z * (kc[1] + z * kc[2] + (z * z) * kc[3]));
388 }
389
390 static const double t0z1 = 0.134861805732790769689793935774652917006,
391 t0z2 = 0.461632144968362341262659542325721328468,
392 t0z3 = 0.819773101100500601787868704921606996312;
393
394 /*
395 * 1.134861805732790769689793935774652917006
396 */
397
398 /*
399 * gamma(x+i) for 0 <= x < 1
400 */
401 static double
402 gam_n(int i, double x)
403 {
404 double rr = 0.0L, yy;
405 double z1, z2;
406
407 /* compute yy = gamma(x+1) */
408 if (x > 0.2845) {
409 if (x > 0.6374)
410 yy = GT3(x - t0z3);
411 else
412 yy = GT2(x - t0z2);
413 } else {
414 yy = GT1(x - t0z1);
415 }
416
417 /* compute gamma(x+i) = (x+i-1)*...*(x+1)*yy, 0<i<8 */
418 switch (i) {
419 case 0: /* yy/x */
420 rr = yy / x;
421 break;
422 case 1: /* yy */
423 rr = yy;
424 break;
425 case 2: /* (x+1)*yy */
426 rr = (x + one) * yy;
427 break;
428 case 3: /* (x+2)*(x+1)*yy */
429 rr = (x + one) * (x + two) * yy;
430 break;
431
432 case 4: /* (x+1)*(x+3)*(x+2)*yy */
433 rr = (x + one) * (x + two) * ((x + 3.0) * yy);
434 break;
435 case 5: /* ((x+1)*(x+4)*(x+2)*(x+3))*yy */
436 z1 = (x + two) * (x + 3.0) * yy;
437 z2 = (x + one) * (x + 4.0);
438 rr = z1 * z2;
439 break;
440 case 6: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5))*yy */
441 z1 = (x + two) * (x + 3.0);
442 z2 = (x + 5.0) * yy;
443 rr = z1 * (z1 - two) * z2;
444 break;
445 case 7: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6))*yy */
446 z1 = (x + two) * (x + 3.0);
447 z2 = (x + 5.0) * (x + 6.0) * yy;
448 rr = z1 * (z1 - two) * z2;
449 break;
450 }
451
452 return (rr);
453 }
454
455 float
456 tgammaf(float xf)
457 {
458 float zf;
459 double ss, ww;
460 double x, y, z;
461 int i, j, k, ix, hx, xk;
462
463 hx = *(int *)&xf;
464 ix = hx & 0x7fffffff;
465
466 x = (double)xf;
467
468 if (ix < 0x33800000)
469 return (1.0F / xf); /* |x| < 2**-24 */
470
471 if (ix >= 0x7f800000)
472 return (xf * ((hx < 0) ? 0.0F : xf)); /* +-Inf or NaN */
473
474 if (hx > 0x420C290F) /* x > 35.040096283... overflow */
475 return ((float)(x / tiny));
476
477 if (hx >= 0x41000000) /* x >= 8 */
478 return ((float)large_gam(x));
479
480 if (hx > 0) { /* 0 < x < 8 */
481 i = (int)xf;
482 return ((float)gam_n(i, x - (double)i));
483 }
484
485 /*
486 * negative x
487 */
488
489 /*
490 * compute xk =
491 * -2 ... x is an even int (-inf is considered even)
492 * -1 ... x is an odd int
493 * +0 ... x is not an int but chopped to an even int
494 * +1 ... x is not an int but chopped to an odd int
495 */
496 xk = 0;
497
498 if (ix >= 0x4b000000) {
499 if (ix > 0x4b000000)
500 xk = -2;
501 else
502 xk = -2 + (ix & 1);
503 } else if (ix >= 0x3f800000) {
504 k = (ix >> 23) - 0x7f;
505 j = ix >> (23 - k);
506
507 if ((j << (23 - k)) == ix)
508 xk = -2 + (j & 1);
509 else
510 xk = j & 1;
511 }
512
513 if (xk < 0) {
514 /* 0/0 invalid NaN, ideally gamma(-n)= (-1)**(n+1) * inf */
515 zf = xf - xf;
516 return (zf / zf);
517 }
518
519 /* negative underflow thresold */
520 if (ix > 0x4224000B) { /* x < -(41+11ulp) */
521 if (xk == 0)
522 z = -tiny;
523 else
524 z = tiny;
525
526 return ((float)z);
527 }
528
529 /*
530 * now compute gamma(x) by -1/((sin(pi*y)/pi)*gamma(1+y)), y = -x
531 */
532
533 /*
534 * First compute ss = -sin(pi*y)/pi , so that
535 * gamma(x) = 1/(ss*gamma(1+y))
536 */
537 y = -x;
538 j = (int)y;
539 z = y - (double)j;
540
541 if (z > 0.3183098861837906715377675) {
542 if (z > 0.6816901138162093284622325)
543 ss = kpsin(one - z);
544 else
545 ss = kpcos(0.5 - z);
546 } else {
547 ss = kpsin(z);
548 }
549
550 if (xk == 0)
551 ss = -ss;
552
553 /* Then compute ww = gamma(1+y) */
554 if (j < 7)
555 ww = gam_n(j + 1, z);
556 else
557 ww = large_gam(y + one);
558
559 /* return 1/(ss*ww) */
560 return ((float)(one / (ww * ss)));
561 }