1 /* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 *
12 */
13 /* lgamma_r(x, signgamp)
14 * Reentrant version of the logarithm of the Gamma function
15 * with user provide pointer for the sign of Gamma(x).
16 *
17 * Method:
18 * 1. Argument Reduction for 0 < x <= 8
19 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
20 * reduce x to a number in [1.5,2.5] by
21 * lgamma(1+s) = log(s) + lgamma(s)
22 * for example,
23 * lgamma(7.3) = log(6.3) + lgamma(6.3)
24 * = log(6.3*5.3) + lgamma(5.3)
25 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
26 * 2. Polynomial approximation of lgamma around its
27 * minimun ymin=1.461632144968362245 to maintain monotonicity.
28 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
29 * Let z = x-ymin;
30 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
31 * where
32 * poly(z) is a 14 degree polynomial.
33 * 2. Rational approximation in the primary interval [2,3]
34 * We use the following approximation:
35 * s = x-2.0;
36 * lgamma(x) = 0.5*s + s*P(s)/Q(s)
37 * with accuracy
38 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
39 * Our algorithms are based on the following observation
40 *
41 * zeta(2)-1 2 zeta(3)-1 3
42 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
43 * 2 3
44 *
45 * where Euler = 0.5771... is the Euler constant, which is very
46 * close to 0.5.
47 *
48 * 3. For x>=8, we have
49 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
50 * (better formula:
51 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
52 * Let z = 1/x, then we approximation
53 * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
54 * by
55 * 3 5 11
56 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
57 * where
58 * |w - f(z)| < 2**-58.74
59 *
60 * 4. For negative x, since (G is gamma function)
61 * -x*G(-x)*G(x) = pi/sin(pi*x),
62 * we have
63 * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
64 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
65 * Hence, for x<0, signgam = sign(sin(pi*x)) and
66 * lgamma(x) = log(|Gamma(x)|)
67 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
68 * Note: one should avoid compute pi*(-x) directly in the
69 * computation of sin(pi*(-x)).
70 *
71 * 5. Special Cases
72 * lgamma(2+s) ~ s*(1-Euler) for tiny s
73 * lgamma(1) = lgamma(2) = 0
74 * lgamma(x) ~ -log(|x|) for tiny x
75 * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
76 * lgamma(inf) = inf
77 * lgamma(-inf) = inf (bug for bug compatible with C99!?)
78 *
79 */
80
81 #include "libm.h"
82
83 static const double
84 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
85 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
86 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
87 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
88 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
89 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
90 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
91 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
92 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
93 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
94 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
95 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
96 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
97 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
98 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
99 /* tt = -(tail of tf) */
100 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
101 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
102 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
103 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
104 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
105 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
106 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
107 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
108 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
109 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
110 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
111 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
112 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
113 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
114 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
115 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
116 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
117 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
118 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
119 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
120 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
121 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
122 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
123 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
124 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
125 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
126 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
127 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
128 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
129 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
130 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
131 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
132 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
133 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
134 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
135 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
136 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
137 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
138 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
139 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
140 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
141 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
142 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
143 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
144 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
145 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
146 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
147
148 /* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
sin_pi(double x)149 static double sin_pi(double x)
150 {
151 int n;
152
153 /* spurious inexact if odd int */
154 x = 2.0*(x*0.5 - floor(x*0.5)); /* x mod 2.0 */
155
156 n = (int)(x*4.0);
157 n = (n+1)/2;
158 x -= n*0.5f;
159 x *= pi;
160
161 switch (n) {
162 default: /* case 4: */
163 case 0: return __sin(x, 0.0, 0);
164 case 1: return __cos(x, 0.0);
165 case 2: return __sin(-x, 0.0, 0);
166 case 3: return -__cos(x, 0.0);
167 }
168 }
169
__lgamma_r(double x,int * signgamp)170 double __lgamma_r(double x, int *signgamp)
171 {
172 union {double f; uint64_t i;} u = {x};
173 double_t t,y,z,nadj,p,p1,p2,p3,q,r,w;
174 uint32_t ix;
175 int sign,i;
176
177 /* purge off +-inf, NaN, +-0, tiny and negative arguments */
178 *signgamp = 1;
179 sign = u.i>>63;
180 ix = u.i>>32 & 0x7fffffff;
181 if (ix >= 0x7ff00000)
182 return x*x;
183 if (ix < (0x3ff-70)<<20) { /* |x|<2**-70, return -log(|x|) */
184 if(sign) {
185 x = -x;
186 *signgamp = -1;
187 }
188 return -log(x);
189 }
190 if (sign) {
191 x = -x;
192 t = sin_pi(x);
193 if (t == 0.0) /* -integer */
194 return 1.0/(x-x);
195 if (t > 0.0)
196 *signgamp = -1;
197 else
198 t = -t;
199 nadj = log(pi/(t*x));
200 }
201
202 /* purge off 1 and 2 */
203 if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t)u.i == 0)
204 r = 0;
205 /* for x < 2.0 */
206 else if (ix < 0x40000000) {
207 if (ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
208 r = -log(x);
209 if (ix >= 0x3FE76944) {
210 y = 1.0 - x;
211 i = 0;
212 } else if (ix >= 0x3FCDA661) {
213 y = x - (tc-1.0);
214 i = 1;
215 } else {
216 y = x;
217 i = 2;
218 }
219 } else {
220 r = 0.0;
221 if (ix >= 0x3FFBB4C3) { /* [1.7316,2] */
222 y = 2.0 - x;
223 i = 0;
224 } else if(ix >= 0x3FF3B4C4) { /* [1.23,1.73] */
225 y = x - tc;
226 i = 1;
227 } else {
228 y = x - 1.0;
229 i = 2;
230 }
231 }
232 switch (i) {
233 case 0:
234 z = y*y;
235 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
236 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
237 p = y*p1+p2;
238 r += (p-0.5*y);
239 break;
240 case 1:
241 z = y*y;
242 w = z*y;
243 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
244 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
245 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
246 p = z*p1-(tt-w*(p2+y*p3));
247 r += tf + p;
248 break;
249 case 2:
250 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
251 p2 = 1.0+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
252 r += -0.5*y + p1/p2;
253 }
254 } else if (ix < 0x40200000) { /* x < 8.0 */
255 i = (int)x;
256 y = x - (double)i;
257 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
258 q = 1.0+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
259 r = 0.5*y+p/q;
260 z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */
261 switch (i) {
262 case 7: z *= y + 6.0; /* FALLTHRU */
263 case 6: z *= y + 5.0; /* FALLTHRU */
264 case 5: z *= y + 4.0; /* FALLTHRU */
265 case 4: z *= y + 3.0; /* FALLTHRU */
266 case 3: z *= y + 2.0; /* FALLTHRU */
267 r += log(z);
268 break;
269 }
270 } else if (ix < 0x43900000) { /* 8.0 <= x < 2**58 */
271 t = log(x);
272 z = 1.0/x;
273 y = z*z;
274 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
275 r = (x-0.5)*(t-1.0)+w;
276 } else /* 2**58 <= x <= inf */
277 r = x*(log(x)-1.0);
278 if (sign)
279 r = nadj - r;
280 return r;
281 }
282
283 weak_alias(__lgamma_r, lgamma_r);
284