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/* Copyright (C) 1994 DJ Delorie, see COPYING.DJ for details */
2
/* @(#)er_lgamma.c 5.1 93/09/24 */
3
/*
4
 * ====================================================
5
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6
 *
7
 * Developed at SunPro, a Sun Microsystems, Inc. business.
8
 * Permission to use, copy, modify, and distribute this
9
 * software is freely granted, provided that this notice
10
 * is preserved.
11
 * ====================================================
12
 */
13
 
14
#if defined(LIBM_SCCS) && !defined(lint)
15
static char rcsid[] = "$Id: e_lgamma_r.c,v 1.5 1994/08/10 20:31:07 jtc Exp $";
16
#endif
17
 
18
/* __ieee754_lgamma_r(x, signgamp)
19
 * Reentrant version of the logarithm of the Gamma function
20
 * with user provide pointer for the sign of Gamma(x).
21
 *
22
 * Method:
23
 *   1. Argument Reduction for 0 < x <= 8
24
 * 	Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
25
 * 	reduce x to a number in [1.5,2.5] by
26
 * 		lgamma(1+s) = log(s) + lgamma(s)
27
 *	for example,
28
 *		lgamma(7.3) = log(6.3) + lgamma(6.3)
29
 *			    = log(6.3*5.3) + lgamma(5.3)
30
 *			    = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
31
 *   2. Polynomial approximation of lgamma around its
32
 *	minimun ymin=1.461632144968362245 to maintain monotonicity.
33
 *	On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
34
 *		Let z = x-ymin;
35
 *		lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
36
 *	where
37
 *		poly(z) is a 14 degree polynomial.
38
 *   2. Rational approximation in the primary interval [2,3]
39
 *	We use the following approximation:
40
 *		s = x-2.0;
41
 *		lgamma(x) = 0.5*s + s*P(s)/Q(s)
42
 *	with accuracy
43
 *		|P/Q - (lgamma(x)-0.5s)| < 2**-61.71
44
 *	Our algorithms are based on the following observation
45
 *
46
 *                             zeta(2)-1    2    zeta(3)-1    3
47
 * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
48
 *                                 2                 3
49
 *
50
 *	where Euler = 0.5771... is the Euler constant, which is very
51
 *	close to 0.5.
52
 *
53
 *   3. For x>=8, we have
54
 *	lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
55
 *	(better formula:
56
 *	   lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
57
 *	Let z = 1/x, then we approximation
58
 *		f(z) = lgamma(x) - (x-0.5)(log(x)-1)
59
 *	by
60
 *	  			    3       5             11
61
 *		w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
62
 *	where
63
 *		|w - f(z)| < 2**-58.74
64
 *
65
 *   4. For negative x, since (G is gamma function)
66
 *		-x*G(-x)*G(x) = pi/sin(pi*x),
67
 * 	we have
68
 * 		G(x) = pi/(sin(pi*x)*(-x)*G(-x))
69
 *	since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
70
 *	Hence, for x<0, signgam = sign(sin(pi*x)) and
71
 *		lgamma(x) = log(|Gamma(x)|)
72
 *			  = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
73
 *	Note: one should avoid compute pi*(-x) directly in the
74
 *	      computation of sin(pi*(-x)).
75
 *
76
 *   5. Special Cases
77
 *		lgamma(2+s) ~ s*(1-Euler) for tiny s
78
 *		lgamma(1)=lgamma(2)=0
79
 *		lgamma(x) ~ -log(x) for tiny x
80
 *		lgamma(0) = lgamma(inf) = inf
81
 *	 	lgamma(-integer) = +-inf
82
 *
83
 */
84
 
85
#include "math.h"
86
#include "math_private.h"
87
 
88
#ifdef __STDC__
89
static const double
90
#else
91
static double
92
#endif
93
two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
94
half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
95
one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
96
pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
97
a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
98
a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
99
a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
100
a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
101
a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
102
a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
103
a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
104
a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
105
a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
106
a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
107
a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
108
a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
109
tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
110
tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
111
/* tt = -(tail of tf) */
112
tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
113
t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
114
t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
115
t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
116
t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
117
t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
118
t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
119
t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
120
t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
121
t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
122
t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
123
t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
124
t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
125
t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
126
t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
127
t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
128
u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
129
u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
130
u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
131
u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
132
u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
133
u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
134
v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
135
v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
136
v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
137
v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
138
v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
139
s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
140
s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
141
s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
142
s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
143
s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
144
s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
145
s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
146
r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
147
r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
148
r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
149
r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
150
r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
151
r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
152
w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
153
w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
154
w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
155
w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
156
w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
157
w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
158
w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
159
 
160
#ifdef __STDC__
161
static const double zero=  0.00000000000000000000e+00;
162
#else
163
static double zero=  0.00000000000000000000e+00;
164
#endif
165
 
166
#ifdef __STDC__
167
	static double sin_pi(double x)
168
#else
169
	static double sin_pi(x)
170
	double x;
171
#endif
172
{
173
	double y,z;
174
	int n,ix;
175
 
176
	GET_HIGH_WORD(ix,x);
177
	ix &= 0x7fffffff;
178
 
179
	if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
180
	y = -x;		/* x is assume negative */
181
 
182
    /*
183
     * argument reduction, make sure inexact flag not raised if input
184
     * is an integer
185
     */
186
	z = floor(y);
187
	if(z!=y) {				/* inexact anyway */
188
	    y  *= 0.5;
189
	    y   = 2.0*(y - floor(y));		/* y = |x| mod 2.0 */
190
	    n   = (int) (y*4.0);
191
	} else {
192
            if(ix>=0x43400000) {
193
                y = zero; n = 0;                 /* y must be even */
194
            } else {
195
                if(ix<0x43300000) z = y+two52;	/* exact */
196
		GET_LOW_WORD(n,z);
197
		n &= 1;
198
                y  = n;
199
                n<<= 2;
200
            }
201
        }
202
	switch (n) {
203
	    case 0:   y =  __kernel_sin(pi*y,zero,0); break;
204
	    case 1:
205
	    case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
206
	    case 3:
207
	    case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
208
	    case 5:
209
	    case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
210
	    default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
211
	    }
212
	return -y;
213
}
214
 
215
 
216
#ifdef __STDC__
217
	double __ieee754_lgamma_r(double x, int *signgamp)
218
#else
219
	double __ieee754_lgamma_r(x,signgamp)
220
	double x; int *signgamp;
221
#endif
222
{
223
	double t,y,z,nadj,p,p1,p2,p3,q,r,w;
224
	int i,hx,lx,ix;
225
 
226
	EXTRACT_WORDS(hx,lx,x);
227
 
228
    /* purge off +-inf, NaN, +-0, and negative arguments */
229
	*signgamp = 1;
230
	ix = hx&0x7fffffff;
231
	if(ix>=0x7ff00000) return x*x;
232
	if((ix|lx)==0) return one/zero;
233
	if(ix<0x3b900000) {	/* |x|<2**-70, return -log(|x|) */
234
	    if(hx<0) {
235
	        *signgamp = -1;
236
	        return -__ieee754_log(-x);
237
	    } else return -__ieee754_log(x);
238
	}
239
	if(hx<0) {
240
	    if(ix>=0x43300000) 	/* |x|>=2**52, must be -integer */
241
		return one/zero;
242
	    t = sin_pi(x);
243
	    if(t==zero) return one/zero; /* -integer */
244
	    nadj = __ieee754_log(pi/fabs(t*x));
245
	    if(t
246
	    x = -x;
247
	}
248
 
249
    /* purge off 1 and 2 */
250
	if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
251
    /* for x < 2.0 */
252
	else if(ix<0x40000000) {
253
	    if(ix<=0x3feccccc) { 	/* lgamma(x) = lgamma(x+1)-log(x) */
254
		r = -__ieee754_log(x);
255
		if(ix>=0x3FE76944) {y = one-x; i= 0;}
256
		else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
257
	  	else {y = x; i=2;}
258
	    } else {
259
	  	r = zero;
260
	        if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
261
	        else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
262
		else {y=x-one;i=2;}
263
	    }
264
	    switch(i) {
265
	      case 0:
266
		z = y*y;
267
		p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
268
		p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
269
		p  = y*p1+p2;
270
		r  += (p-0.5*y); break;
271
	      case 1:
272
		z = y*y;
273
		w = z*y;
274
		p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));	/* parallel comp */
275
		p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
276
		p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
277
		p  = z*p1-(tt-w*(p2+y*p3));
278
		r += (tf + p); break;
279
	      case 2:
280
		p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
281
		p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
282
		r += (-0.5*y + p1/p2);
283
	    }
284
	}
285
	else if(ix<0x40200000) { 			/* x < 8.0 */
286
	    i = (int)x;
287
	    t = zero;
288
	    y = x-(double)i;
289
	    p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
290
	    q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
291
	    r = half*y+p/q;
292
	    z = one;	/* lgamma(1+s) = log(s) + lgamma(s) */
293
	    switch(i) {
294
	    case 7: z *= (y+6.0);	/* FALLTHRU */
295
	    case 6: z *= (y+5.0);	/* FALLTHRU */
296
	    case 5: z *= (y+4.0);	/* FALLTHRU */
297
	    case 4: z *= (y+3.0);	/* FALLTHRU */
298
	    case 3: z *= (y+2.0);	/* FALLTHRU */
299
		    r += __ieee754_log(z); break;
300
	    }
301
    /* 8.0 <= x < 2**58 */
302
	} else if (ix < 0x43900000) {
303
	    t = __ieee754_log(x);
304
	    z = one/x;
305
	    y = z*z;
306
	    w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
307
	    r = (x-half)*(t-one)+w;
308
	} else
309
    /* 2**58 <= x <= inf */
310
	    r =  x*(__ieee754_log(x)-one);
311
	if(hx<0) r = nadj - r;
312
	return r;
313
}