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 <math.h>
44#if defined(__SUNPRO_C)
45#include <sunmath.h>
46#endif
47#include <sys/isa_defs.h>
48
49#if defined(_BIG_ENDIAN)
50#define	HIWORD	0
51#define	LOWORD	1
52#else
53#define	HIWORD	1
54#define	LOWORD	0
55#endif
56#define	__HI(x)	((int *) &x)[HIWORD]
57#define	__LO(x)	((unsigned *) &x)[LOWORD]
58
59/* Coefficients for primary intervals GTi() */
60static const double cr[] = {
61	/* p1 */
62	+7.09087253435088360271451613398019280077561279443e-0001,
63	-5.17229560788652108545141978238701790105241761089e-0001,
64	+5.23403394528150789405825222323770647162337764327e-0001,
65	-4.54586308717075010784041566069480411732634814899e-0001,
66	+4.20596490915239085459964590559256913498190955233e-0001,
67	-3.57307589712377520978332185838241458642142185789e-0001,
68
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
77	/* p3 */
78	+3.82409531118807759081121479786092134814808872880e-0001,
79	+2.65309888180188647956400403013495759365167853426e-0002,
80	+8.06815109775079171923561169415370309376296739835e-0002,
81	-1.54821591666137613928840890835174351674007764799e-0002,
82	+1.76308239242717268530498313416899188157165183405e-0002,
83
84	/* GZi and TZi */
85	+0.9382046279096824494097535615803269576988,	/* GZ1 */
86	+0.8856031944108887002788159005825887332080,	/* GZ2 */
87	+0.9367814114636523216188468970808378497426,	/* GZ3 */
88	-0.3517214357852935791015625,	/* TZ1 */
89	+0.280530631542205810546875,	/* TZ3 */
90};
91
92#define	P10	cr[0]
93#define	P11	cr[1]
94#define	P12	cr[2]
95#define	P13	cr[3]
96#define	P14	cr[4]
97#define	P15	cr[5]
98#define	P20	cr[6]
99#define	P21	cr[7]
100#define	P22	cr[8]
101#define	P23	cr[9]
102#define	P24	cr[10]
103#define	P25	cr[11]
104#define	P30	cr[12]
105#define	P31	cr[13]
106#define	P32	cr[14]
107#define	P33	cr[15]
108#define	P34	cr[16]
109#define	GZ1	cr[17]
110#define	GZ2	cr[18]
111#define	GZ3	cr[19]
112#define	TZ1	cr[20]
113#define	TZ3	cr[21]
114
115/* compute gamma(y) for y in GT1 = [1.0000, 1.2845] */
116static double
117GT1(double y) {
118	double z, r;
119
120	z = y * y;
121	r = TZ1 * y + z * ((P10 + y * P11 + z * P12) + (z * y) * (P13 + y *
122		P14 + z * P15));
123	return (GZ1 + r);
124}
125
126/* compute gamma(y) for y in GT2 = [1.2844, 1.6374] */
127static double
128GT2(double y) {
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] */
137static double
138GT3(double y) {
139double z, r;
140
141	z = y * y;
142	r = TZ3 * y + z * ((P30 + y * P31 + z * P32) + (z * y) * (P33 + y *
143		P34));
144	return (GZ3 + r);
145}
146
147/* INDENT OFF */
148static 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) */
179static 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/* INDENT ON */
214
215/* INDENT OFF */
216/*
217 * return tgammaf(x) in double for 8<x<=35.040096283... using Stirling's formula
218 *     log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
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
233static const double T1[] = { /* T1[j]=(j+3)*log(2)-1 */
234+1.079441541679835928251696364375e+00,
235+1.772588722239781237668928485833e+00,
236+2.465735902799726547086160607291e+00,
237};
238
239static const double T2[] = {   /* T2[j]=log(1+j/64+1/128) */
240+7.782140442054948947462900061137e-03,
241+2.316705928153437822879916096229e-02,
242+3.831886430213659919375532512380e-02,
243+5.324451451881228286587019378653e-02,
244+6.795066190850774939456527777263e-02,
245+8.244366921107459126816006866831e-02,
246+9.672962645855111229557105648746e-02,
247+1.108143663402901141948061693232e-01,
248+1.247034785009572358634065153809e-01,
249+1.384023228591191356853258736016e-01,
250+1.519160420258419750718034248969e-01,
251+1.652495728953071628756114492772e-01,
252+1.784076574728182971194002415109e-01,
253+1.913948529996294546092988075613e-01,
254+2.042155414286908915038203861962e-01,
255+2.168739383006143596190895257443e-01,
256+2.293741010648458299914807250461e-01,
257+2.417199368871451681443075159135e-01,
258+2.539152099809634441373232979066e-01,
259+2.659635484971379413391259265375e-01,
260+2.778684510034563061863500329234e-01,
261+2.896332925830426768788930555257e-01,
262+3.012613305781617810128755382338e-01,
263+3.127557100038968883862465596883e-01,
264+3.241194686542119760906707604350e-01,
265+3.353555419211378302571795798142e-01,
266+3.464667673462085809184621884258e-01,
267+3.574558889218037742260094901409e-01,
268+3.683255611587076530482301540504e-01,
269+3.790783529349694583908533456310e-01,
270+3.897167511400252133704636040035e-01,
271+4.002431641270127069293251019951e-01,
272+4.106599249852683859343062031758e-01,
273+4.209692946441296361288671615068e-01,
274+4.311734648183713408591724789556e-01,
275+4.412745608048752294894964416613e-01,
276+4.512746441394585851446923830790e-01,
277+4.611757151221701663679999255979e-01,
278+4.709797152187910125468978560564e-01,
279+4.806885293457519076766184554480e-01,
280+4.903039880451938381503461596457e-01,
281+4.998278695564493298213314152470e-01,
282+5.092619017898079468040749192283e-01,
283+5.186077642080456321529769963648e-01,
284+5.278670896208423851138922177783e-01,
285+5.370414658968836545667292441538e-01,
286+5.461324375981356503823972092312e-01,
287+5.551415075405015927154803595159e-01,
288+5.640701382848029660713842900902e-01,
289+5.729197535617855090927567266263e-01,
290+5.816917396346224825206107537254e-01,
291+5.903874466021763746419167081236e-01,
292+5.990081896460833993816000244617e-01,
293+6.075552502245417955010851527911e-01,
294+6.160298772155140196475659281967e-01,
295+6.244332880118935010425387440547e-01,
296+6.327666695710378295457864685036e-01,
297+6.410311794209312910556013344054e-01,
298+6.492279466251098188908399699053e-01,
299+6.573580727083600301418900232459e-01,
300+6.654226325450904489500926100067e-01,
301+6.734226752121667202979603888010e-01,
302+6.813592248079030689480715595681e-01,
303+6.892332812388089803249143378146e-01,
304};
305/* INDENT ON */
306
307static double
308large_gam(double x) {
309	double ss, zz, z, t1, t2, w, y, u;
310	unsigned lx;
311	int k, ix, j, m;
312
313	ix = __HI(x);
314	lx = __LO(x);
315	m = (ix >> 20) - 0x3ff;			/* exponent of x, range:3-5 */
316	ix = (ix & 0x000fffff) | 0x3ff00000;	/* y = scale x to [1,2] */
317	__HI(y) = ix;
318	__LO(y) = lx;
319	__HI(z) = (ix & 0xffffc000) | 0x2000;	/* z[j]=1+j/64+1/128 */
320	__LO(z) = 0;
321	j = (ix >> 14) & 0x3f;
322	t1 = y + z;
323	t2 = y - z;
324	u = t2 / t1;
325	ss = T1[m - 3] + T2[j] + u * (two + A1 * (u * u));
326							/* ss = log(x)-1 */
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/* INDENT OFF */
345/*
346 * kpsin(x)= sin(pi*x)/pi
347 *                 3        5        7        9
348 *	= x+ks[0]*x +ks[1]*x +ks[2]*x +ks[3]*x
349 */
350static const double ks[] = {
351-1.64493404985645811354476665052005342839447790544e+0000,
352+8.11740794458351064092797249069438269367389272270e-0001,
353-1.90703144603551216933075809162889536878854055202e-0001,
354+2.55742333994264563281155312271481108635575331201e-0002,
355};
356/* INDENT ON */
357
358static double
359kpsin(double x) {
360	double z;
361
362	z = x * x;
363	return (x + (x * z) * ((ks[0] + z * ks[1]) + (z * z) * (ks[2] + z *
364		ks[3])));
365}
366
367/* INDENT OFF */
368/*
369 * kpcos(x)= cos(pi*x)/pi
370 *                     2        4        6
371 *	= kc[0]+kc[1]*x +kc[2]*x +kc[3]*x
372 */
373static const double kc[] = {
374+3.18309886183790671537767526745028724068919291480e-0001,
375-1.57079581447762568199467875065854538626594937791e+0000,
376+1.29183528092558692844073004029568674027807393862e+0000,
377-4.20232949771307685981015914425195471602739075537e-0001,
378};
379/* INDENT ON */
380
381static double
382kpcos(double x) {
383	double z;
384
385	z = x * x;
386	return (kc[0] + z * (kc[1] + z * kc[2] + (z * z) * kc[3]));
387}
388
389/* INDENT OFF */
390static const double
391t0z1 = 0.134861805732790769689793935774652917006,
392t0z2 = 0.461632144968362341262659542325721328468,
393t0z3 = 0.819773101100500601787868704921606996312;
394	/* 1.134861805732790769689793935774652917006 */
395/* INDENT ON */
396
397/*
398 * gamma(x+i) for 0 <= x < 1
399 */
400static double
401gam_n(int i, double x) {
402	double rr = 0.0L, yy;
403	double z1, z2;
404
405	/* compute yy = gamma(x+1) */
406	if (x > 0.2845) {
407		if (x > 0.6374)
408			yy = GT3(x - t0z3);
409		else
410			yy = GT2(x - t0z2);
411	} else
412		yy = GT1(x - t0z1);
413
414	/* compute gamma(x+i) = (x+i-1)*...*(x+1)*yy, 0<i<8 */
415	switch (i) {
416	case 0:		/* yy/x */
417		rr = yy / x;
418		break;
419	case 1:		/* yy */
420		rr = yy;
421		break;
422	case 2:		/* (x+1)*yy */
423		rr = (x + one) * yy;
424		break;
425	case 3:		/* (x+2)*(x+1)*yy */
426		rr = (x + one) * (x + two) * yy;
427		break;
428
429	case 4:		/* (x+1)*(x+3)*(x+2)*yy */
430		rr = (x + one) * (x + two) * ((x + 3.0) * yy);
431		break;
432	case 5:		/* ((x+1)*(x+4)*(x+2)*(x+3))*yy */
433		z1 = (x + two) * (x + 3.0) * yy;
434		z2 = (x + one) * (x + 4.0);
435		rr = z1 * z2;
436		break;
437	case 6:		/* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5))*yy */
438		z1 = (x + two) * (x + 3.0);
439		z2 = (x + 5.0) * yy;
440		rr = z1 * (z1 - two) * z2;
441		break;
442	case 7:		/* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6))*yy */
443		z1 = (x + two) * (x + 3.0);
444		z2 = (x + 5.0) * (x + 6.0) * yy;
445		rr = z1 * (z1 - two) * z2;
446		break;
447	}
448	return (rr);
449}
450
451float
452tgammaf(float xf) {
453	float zf;
454	double ss, ww;
455	double x, y, z;
456	int i, j, k, ix, hx, xk;
457
458	hx = *(int *) &xf;
459	ix = hx & 0x7fffffff;
460
461	x = (double) xf;
462	if (ix < 0x33800000)
463		return (1.0F / xf);	/* |x| < 2**-24 */
464
465	if (ix >= 0x7f800000)
466		return (xf * ((hx < 0)? 0.0F : xf)); /* +-Inf or NaN */
467
468	if (hx > 0x420C290F) 	/* x > 35.040096283... overflow */
469		return (float)(x / tiny);
470
471	if (hx >= 0x41000000)	/* x >= 8 */
472		return ((float) large_gam(x));
473
474	if (hx > 0) {		/* 0 < x < 8 */
475		i = (int) xf;
476		return ((float) gam_n(i, x - (double) i));
477	}
478
479	/* negative x */
480	/* INDENT OFF */
481	/*
482	 * compute xk =
483	 *	-2 ... x is an even int (-inf is considered even)
484	 *	-1 ... x is an odd int
485	 *	+0 ... x is not an int but chopped to an even int
486	 *	+1 ... x is not an int but chopped to an odd int
487	 */
488	/* INDENT ON */
489	xk = 0;
490	if (ix >= 0x4b000000) {
491		if (ix > 0x4b000000)
492			xk = -2;
493		else
494			xk = -2 + (ix & 1);
495	} else if (ix >= 0x3f800000) {
496		k = (ix >> 23) - 0x7f;
497		j = ix >> (23 - k);
498		if ((j << (23 - k)) == ix)
499			xk = -2 + (j & 1);
500		else
501			xk = j & 1;
502	}
503	if (xk < 0) {
504		/* 0/0 invalid NaN, ideally gamma(-n)= (-1)**(n+1) * inf */
505		zf = xf - xf;
506		return (zf / zf);
507	}
508
509	/* negative underflow thresold */
510	if (ix > 0x4224000B) {	/* x < -(41+11ulp) */
511		if (xk == 0)
512			z = -tiny;
513		else
514			z = tiny;
515		return ((float)z);
516	}
517
518	/* INDENT OFF */
519	/* now compute gamma(x) by  -1/((sin(pi*y)/pi)*gamma(1+y)), y = -x */
520	/*
521	 * First compute ss = -sin(pi*y)/pi , so that
522	 * gamma(x) = 1/(ss*gamma(1+y))
523	 */
524	/* INDENT ON */
525	y = -x;
526	j = (int) y;
527	z = y - (double) j;
528	if (z > 0.3183098861837906715377675)
529		if (z > 0.6816901138162093284622325)
530			ss = kpsin(one - z);
531		else
532			ss = kpcos(0.5 - z);
533	else
534		ss = kpsin(z);
535	if (xk == 0)
536		ss = -ss;
537
538	/* Then compute ww = gamma(1+y)  */
539	if (j < 7)
540		ww = gam_n(j + 1, z);
541	else
542		ww = large_gam(y + one);
543
544	/* return 1/(ss*ww) */
545	return ((float) (one / (ww * ss)));
546}
547