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