1/*
2"A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964)
3"Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001)
4"An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004)
5
6approximation method:
7
8                        (x - 0.5)         S(x)
9Gamma(x) = (x + g - 0.5)         *  ----------------
10                                    exp(x + g - 0.5)
11
12with
13                 a1      a2      a3            aN
14S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ]
15               x + 1   x + 2   x + 3         x + N
16
17with a0, a1, a2, a3,.. aN constants which depend on g.
18
19for x < 0 the following reflection formula is used:
20
21Gamma(x)*Gamma(-x) = -pi/(x sin(pi x))
22
23most ideas and constants are from boost and python
24*/
25#include "libm.h"
26
27static const double pi = 3.141592653589793238462643383279502884;
28
29/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */
30static double sinpi(double x)
31{
32	int n;
33
34	/* argument reduction: x = |x| mod 2 */
35	/* spurious inexact when x is odd int */
36	x = x * 0.5;
37	x = 2 * (x - floor(x));
38
39	/* reduce x into [-.25,.25] */
40	n = 4 * x;
41	n = (n+1)/2;
42	x -= n * 0.5;
43
44	x *= pi;
45	switch (n) {
46	default: /* case 4 */
47	case 0:
48		return __sin(x, 0, 0);
49	case 1:
50		return __cos(x, 0);
51	case 2:
52		return __sin(-x, 0, 0);
53	case 3:
54		return -__cos(x, 0);
55	}
56}
57
58#define N 12
59//static const double g = 6.024680040776729583740234375;
60static const double gmhalf = 5.524680040776729583740234375;
61static const double Snum[N+1] = {
62	23531376880.410759688572007674451636754734846804940,
63	42919803642.649098768957899047001988850926355848959,
64	35711959237.355668049440185451547166705960488635843,
65	17921034426.037209699919755754458931112671403265390,
66	6039542586.3520280050642916443072979210699388420708,
67	1439720407.3117216736632230727949123939715485786772,
68	248874557.86205415651146038641322942321632125127801,
69	31426415.585400194380614231628318205362874684987640,
70	2876370.6289353724412254090516208496135991145378768,
71	186056.26539522349504029498971604569928220784236328,
72	8071.6720023658162106380029022722506138218516325024,
73	210.82427775157934587250973392071336271166969580291,
74	2.5066282746310002701649081771338373386264310793408,
75};
76static const double Sden[N+1] = {
77	0, 39916800, 120543840, 150917976, 105258076, 45995730, 13339535,
78	2637558, 357423, 32670, 1925, 66, 1,
79};
80/* n! for small integer n */
81static const double fact[] = {
82	1, 1, 2, 6, 24, 120, 720, 5040.0, 40320.0, 362880.0, 3628800.0, 39916800.0,
83	479001600.0, 6227020800.0, 87178291200.0, 1307674368000.0, 20922789888000.0,
84	355687428096000.0, 6402373705728000.0, 121645100408832000.0,
85	2432902008176640000.0, 51090942171709440000.0, 1124000727777607680000.0,
86};
87
88/* S(x) rational function for positive x */
89static double S(double x)
90{
91	double_t num = 0, den = 0;
92	int i;
93
94	/* to avoid overflow handle large x differently */
95	if (x < 8)
96		for (i = N; i >= 0; i--) {
97			num = num * x + Snum[i];
98			den = den * x + Sden[i];
99		}
100	else
101		for (i = 0; i <= N; i++) {
102			num = num / x + Snum[i];
103			den = den / x + Sden[i];
104		}
105	return num/den;
106}
107
108double tgamma(double x)
109{
110	union {double f; uint64_t i;} u = {x};
111	double absx, y;
112	double_t dy, z, r;
113	uint32_t ix = u.i>>32 & 0x7fffffff;
114	int sign = u.i>>63;
115
116	/* special cases */
117	if (ix >= 0x7ff00000)
118		/* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */
119		return x + INFINITY;
120	if (ix < (0x3ff-54)<<20)
121		/* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */
122		return 1/x;
123
124	/* integer arguments */
125	/* raise inexact when non-integer */
126	if (x == floor(x)) {
127		if (sign)
128			return 0/0.0;
129		if (x <= sizeof fact/sizeof *fact)
130			return fact[(int)x - 1];
131	}
132
133	/* x >= 172: tgamma(x)=inf with overflow */
134	/* x =< -184: tgamma(x)=+-0 with underflow */
135	if (ix >= 0x40670000) { /* |x| >= 184 */
136		if (sign) {
137			FORCE_EVAL((float)(0x1p-126/x));
138			if (floor(x) * 0.5 == floor(x * 0.5))
139				return 0;
140			return -0.0;
141		}
142		x *= 0x1p1023;
143		return x;
144	}
145
146	absx = sign ? -x : x;
147
148	/* handle the error of x + g - 0.5 */
149	y = absx + gmhalf;
150	if (absx > gmhalf) {
151		dy = y - absx;
152		dy -= gmhalf;
153	} else {
154		dy = y - gmhalf;
155		dy -= absx;
156	}
157
158	z = absx - 0.5;
159	r = S(absx) * exp(-y);
160	if (x < 0) {
161		/* reflection formula for negative x */
162		/* sinpi(absx) is not 0, integers are already handled */
163		r = -pi / (sinpi(absx) * absx * r);
164		dy = -dy;
165		z = -z;
166	}
167	r += dy * (gmhalf+0.5) * r / y;
168	z = pow(y, 0.5*z);
169	y = r * z * z;
170	return y;
171}
172
173#if 0
174double __lgamma_r(double x, int *sign)
175{
176	double r, absx;
177
178	*sign = 1;
179
180	/* special cases */
181	if (!isfinite(x))
182		/* lgamma(nan)=nan, lgamma(+-inf)=inf */
183		return x*x;
184
185	/* integer arguments */
186	if (x == floor(x) && x <= 2) {
187		/* n <= 0: lgamma(n)=inf with divbyzero */
188		/* n == 1,2: lgamma(n)=0 */
189		if (x <= 0)
190			return 1/0.0;
191		return 0;
192	}
193
194	absx = fabs(x);
195
196	/* lgamma(x) ~ -log(|x|) for tiny |x| */
197	if (absx < 0x1p-54) {
198		*sign = 1 - 2*!!signbit(x);
199		return -log(absx);
200	}
201
202	/* use tgamma for smaller |x| */
203	if (absx < 128) {
204		x = tgamma(x);
205		*sign = 1 - 2*!!signbit(x);
206		return log(fabs(x));
207	}
208
209	/* second term (log(S)-g) could be more precise here.. */
210	/* or with stirling: (|x|-0.5)*(log(|x|)-1) + poly(1/|x|) */
211	r = (absx-0.5)*(log(absx+gmhalf)-1) + (log(S(absx)) - (gmhalf+0.5));
212	if (x < 0) {
213		/* reflection formula for negative x */
214		x = sinpi(absx);
215		*sign = 2*!!signbit(x) - 1;
216		r = log(pi/(fabs(x)*absx)) - r;
217	}
218	return r;
219}
220
221weak_alias(__lgamma_r, lgamma_r);
222#endif
223