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e_lgamma_r.c (10941B)
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1 /* @(#)er_lgamma.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, 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 #ifndef lint
14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_lgamma_r.c,v 1.7 2002/05/28 18:15:04 alfred Exp $";
15 #endif
16
17 /* __ieee754_lgamma_r(x, signgamp)
18 * Reentrant version of the logarithm of the Gamma function
19 * with user provide pointer for the sign of Gamma(x).
20 *
21 * Method:
22 * 1. Argument Reduction for 0 < x <= 8
23 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
24 * reduce x to a number in [1.5,2.5] by
25 * lgamma(1+s) = log(s) + lgamma(s)
26 * for example,
27 * lgamma(7.3) = log(6.3) + lgamma(6.3)
28 * = log(6.3*5.3) + lgamma(5.3)
29 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
30 * 2. Polynomial approximation of lgamma around its
31 * minimun ymin=1.461632144968362245 to maintain monotonicity.
32 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
33 * Let z = x-ymin;
34 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
35 * where
36 * poly(z) is a 14 degree polynomial.
37 * 2. Rational approximation in the primary interval [2,3]
38 * We use the following approximation:
39 * s = x-2.0;
40 * lgamma(x) = 0.5*s + s*P(s)/Q(s)
41 * with accuracy
42 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
43 * Our algorithms are based on the following observation
44 *
45 * zeta(2)-1 2 zeta(3)-1 3
46 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
47 * 2 3
48 *
49 * where Euler = 0.5771... is the Euler constant, which is very
50 * close to 0.5.
51 *
52 * 3. For x>=8, we have
53 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
54 * (better formula:
55 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
56 * Let z = 1/x, then we approximation
57 * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
58 * by
59 * 3 5 11
60 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
61 * where
62 * |w - f(z)| < 2**-58.74
63 *
64 * 4. For negative x, since (G is gamma function)
65 * -x*G(-x)*G(x) = pi/sin(pi*x),
66 * we have
67 * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
68 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
69 * Hence, for x<0, signgam = sign(sin(pi*x)) and
70 * lgamma(x) = log(|Gamma(x)|)
71 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
72 * Note: one should avoid compute pi*(-x) directly in the
73 * computation of sin(pi*(-x)).
74 *
75 * 5. Special Cases
76 * lgamma(2+s) ~ s*(1-Euler) for tiny s
77 * lgamma(1)=lgamma(2)=0
78 * lgamma(x) ~ -log(x) for tiny x
79 * lgamma(0) = lgamma(inf) = inf
80 * lgamma(-integer) = +-inf
81 *
82 */
83
84 #include "math.h"
85 #include "math_private.h"
86
87 static const double
88 two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
89 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
90 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
91 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
92 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
93 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
94 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
95 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
96 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
97 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
98 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
99 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
100 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
101 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
102 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
103 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
104 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
105 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
106 /* tt = -(tail of tf) */
107 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
108 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
109 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
110 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
111 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
112 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
113 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
114 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
115 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
116 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
117 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
118 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
119 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
120 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
121 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
122 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
123 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
124 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
125 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
126 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
127 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
128 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
129 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
130 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
131 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
132 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
133 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
134 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
135 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
136 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
137 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
138 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
139 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
140 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
141 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
142 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
143 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
144 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
145 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
146 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
147 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
148 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
149 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
150 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
151 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
152 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
153 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
154
155 static const double zero= 0.00000000000000000000e+00;
156
157 static double sin_pi(double x)
158 {
159 double y,z;
160 int n,ix;
161
162 GET_HIGH_WORD(ix,x);
163 ix &= 0x7fffffff;
164
165 if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
166 y = -x; /* x is assume negative */
167
168 /*
169 * argument reduction, make sure inexact flag not raised if input
170 * is an integer
171 */
172 z = floor(y);
173 if(z!=y) { /* inexact anyway */
174 y *= 0.5;
175 y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */
176 n = (int) (y*4.0);
177 } else {
178 if(ix>=0x43400000) {
179 y = zero; n = 0; /* y must be even */
180 } else {
181 if(ix<0x43300000) z = y+two52; /* exact */
182 GET_LOW_WORD(n,z);
183 n &= 1;
184 y = n;
185 n<<= 2;
186 }
187 }
188 switch (n) {
189 case 0: y = __kernel_sin(pi*y,zero,0); break;
190 case 1:
191 case 2: y = __kernel_cos(pi*(0.5-y),zero); break;
192 case 3:
193 case 4: y = __kernel_sin(pi*(one-y),zero,0); break;
194 case 5:
195 case 6: y = -__kernel_cos(pi*(y-1.5),zero); break;
196 default: y = __kernel_sin(pi*(y-2.0),zero,0); break;
197 }
198 return -y;
199 }
200
201
202 double
203 __ieee754_lgamma_r(double x, int *signgamp)
204 {
205 double t,y,z,nadj,p,p1,p2,p3,q,r,w;
206 int i,hx,lx,ix;
207
208 EXTRACT_WORDS(hx,lx,x);
209
210 /* purge off +-inf, NaN, +-0, and negative arguments */
211 *signgamp = 1;
212 ix = hx&0x7fffffff;
213 if(ix>=0x7ff00000) return x*x;
214 if((ix|lx)==0) return one/zero;
215 if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
216 if(hx<0) {
217 *signgamp = -1;
218 return -__ieee754_log(-x);
219 } else return -__ieee754_log(x);
220 }
221 if(hx<0) {
222 if(ix>=0x43300000) /* |x|>=2**52, must be -integer */
223 return one/zero;
224 t = sin_pi(x);
225 if(t==zero) return one/zero; /* -integer */
226 nadj = __ieee754_log(pi/fabs(t*x));
227 if(t<zero) *signgamp = -1;
228 x = -x;
229 }
230
231 /* purge off 1 and 2 */
232 if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
233 /* for x < 2.0 */
234 else if(ix<0x40000000) {
235 if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
236 r = -__ieee754_log(x);
237 if(ix>=0x3FE76944) {y = one-x; i= 0;}
238 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
239 else {y = x; i=2;}
240 } else {
241 r = zero;
242 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
243 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
244 else {y=x-one;i=2;}
245 }
246 switch(i) {
247 case 0:
248 z = y*y;
249 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
250 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
251 p = y*p1+p2;
252 r += (p-0.5*y); break;
253 case 1:
254 z = y*y;
255 w = z*y;
256 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
257 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
258 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
259 p = z*p1-(tt-w*(p2+y*p3));
260 r += (tf + p); break;
261 case 2:
262 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
263 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
264 r += (-0.5*y + p1/p2);
265 }
266 }
267 else if(ix<0x40200000) { /* x < 8.0 */
268 i = (int)x;
269 t = zero;
270 y = x-(double)i;
271 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
272 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
273 r = half*y+p/q;
274 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
275 switch(i) {
276 case 7: z *= (y+6.0); /* FALLTHRU */
277 case 6: z *= (y+5.0); /* FALLTHRU */
278 case 5: z *= (y+4.0); /* FALLTHRU */
279 case 4: z *= (y+3.0); /* FALLTHRU */
280 case 3: z *= (y+2.0); /* FALLTHRU */
281 r += __ieee754_log(z); break;
282 }
283 /* 8.0 <= x < 2**58 */
284 } else if (ix < 0x43900000) {
285 t = __ieee754_log(x);
286 z = one/x;
287 y = z*z;
288 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
289 r = (x-half)*(t-one)+w;
290 } else
291 /* 2**58 <= x <= inf */
292 r = x*(__ieee754_log(x)-one);
293 if(hx<0) r = nadj - r;
294 return r;
295 }