1
2/* @(#)e_lgamma_r.c 1.3 95/01/18 */
3/*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunSoft, 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
15/* __ieee754_lgamma_r(x, signgamp)
16 * Reentrant version of the logarithm of the Gamma function
17 * with user provide pointer for the sign of Gamma(x).
18 *
19 * Method:
20 *   1. Argument Reduction for 0 < x <= 8
21 * 	Since ieee_gamma(1+s)=s*ieee_gamma(s), for x in [0,8], we may
22 * 	reduce x to a number in [1.5,2.5] by
23 * 		lgamma(1+s) = ieee_log(s) + ieee_lgamma(s)
24 *	for example,
25 *		lgamma(7.3) = ieee_log(6.3) + ieee_lgamma(6.3)
26 *			    = ieee_log(6.3*5.3) + ieee_lgamma(5.3)
27 *			    = ieee_log(6.3*5.3*4.3*3.3*2.3) + ieee_lgamma(2.3)
28 *   2. Polynomial approximation of lgamma around its
29 *	minimun ymin=1.461632144968362245 to maintain monotonicity.
30 *	On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
31 *		Let z = x-ymin;
32 *		lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
33 *	where
34 *		poly(z) is a 14 degree polynomial.
35 *   2. Rational approximation in the primary interval [2,3]
36 *	We use the following approximation:
37 *		s = x-2.0;
38 *		lgamma(x) = 0.5*s + s*P(s)/Q(s)
39 *	with accuracy
40 *		|P/Q - (ieee_lgamma(x)-0.5s)| < 2**-61.71
41 *	Our algorithms are based on the following observation
42 *
43 *                             zeta(2)-1    2    zeta(3)-1    3
44 * ieee_lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
45 *                                 2                 3
46 *
47 *	where Euler = 0.5771... is the Euler constant, which is very
48 *	close to 0.5.
49 *
50 *   3. For x>=8, we have
51 *	lgamma(x)~(x-0.5)log(x)-x+0.5*ieee_log(2pi)+1/(12x)-1/(360x**3)+....
52 *	(better formula:
53 *	   ieee_lgamma(x)~(x-0.5)*(ieee_log(x)-1)-.5*(ieee_log(2pi)-1) + ...)
54 *	Let z = 1/x, then we approximation
55 *		f(z) = ieee_lgamma(x) - (x-0.5)(ieee_log(x)-1)
56 *	by
57 *	  			    3       5             11
58 *		w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
59 *	where
60 *		|w - f(z)| < 2**-58.74
61 *
62 *   4. For negative x, since (G is gamma function)
63 *		-x*G(-x)*G(x) = pi/ieee_sin(pi*x),
64 * 	we have
65 * 		G(x) = pi/(ieee_sin(pi*x)*(-x)*G(-x))
66 *	since G(-x) is positive, sign(G(x)) = sign(ieee_sin(pi*x)) for x<0
67 *	Hence, for x<0, signgam = sign(ieee_sin(pi*x)) and
68 *		lgamma(x) = ieee_log(|Gamma(x)|)
69 *			  = ieee_log(pi/(|x*ieee_sin(pi*x)|)) - ieee_lgamma(-x);
70 *	Note: one should avoid compute pi*(-x) directly in the
71 *	      computation of ieee_sin(pi*(-x)).
72 *
73 *   5. Special Cases
74 *		lgamma(2+s) ~ s*(1-Euler) for tiny s
75 *		lgamma(1)=ieee_lgamma(2)=0
76 *		lgamma(x) ~ -ieee_log(x) for tiny x
77 *		lgamma(0) = ieee_lgamma(inf) = inf
78 *	 	lgamma(-integer) = +-inf
79 *
80 */
81
82#include "fdlibm.h"
83
84#ifdef __STDC__
85static const double
86#else
87static double
88#endif
89two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
90half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
91one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
92pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
93a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
94a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
95a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
96a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
97a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
98a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
99a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
100a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
101a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
102a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
103a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
104a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
105tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
106tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
107/* tt = -(tail of tf) */
108tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
109t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
110t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
111t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
112t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
113t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
114t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
115t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
116t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
117t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
118t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
119t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
120t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
121t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
122t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
123t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
124u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
125u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
126u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
127u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
128u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
129u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
130v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
131v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
132v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
133v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
134v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
135s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
136s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
137s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
138s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
139s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
140s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
141s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
142r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
143r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
144r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
145r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
146r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
147r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
148w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
149w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
150w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
151w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
152w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
153w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
154w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
155
156static double zero=  0.00000000000000000000e+00;
157
158#ifdef __STDC__
159	static double sin_pi(double x)
160#else
161	static double sin_pi(x)
162	double x;
163#endif
164{
165	double y,z;
166	int n,ix;
167
168	ix = 0x7fffffff&__HI(x);
169
170	if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
171	y = -x;		/* x is assume negative */
172
173    /*
174     * argument reduction, make sure inexact flag not raised if input
175     * is an integer
176     */
177	z = ieee_floor(y);
178	if(z!=y) {				/* inexact anyway */
179	    y  *= 0.5;
180	    y   = 2.0*(y - ieee_floor(y));		/* y = |x| mod 2.0 */
181	    n   = (int) (y*4.0);
182	} else {
183            if(ix>=0x43400000) {
184                y = zero; n = 0;                 /* y must be even */
185            } else {
186                if(ix<0x43300000) z = y+two52;	/* exact */
187                n   = __LO(z)&1;        /* lower word of z */
188                y  = n;
189                n<<= 2;
190            }
191        }
192	switch (n) {
193	    case 0:   y =  __kernel_sin(pi*y,zero,0); break;
194	    case 1:
195	    case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
196	    case 3:
197	    case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
198	    case 5:
199	    case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
200	    default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
201	    }
202	return -y;
203}
204
205
206#ifdef __STDC__
207	double __ieee754_lgamma_r(double x, int *signgamp)
208#else
209	double __ieee754_lgamma_r(x,signgamp)
210	double x; int *signgamp;
211#endif
212{
213	double t,y,z,nadj,p,p1,p2,p3,q,r,w;
214	int i,hx,lx,ix;
215
216	hx = __HI(x);
217	lx = __LO(x);
218
219    /* purge off +-inf, NaN, +-0, and negative arguments */
220	*signgamp = 1;
221	ix = hx&0x7fffffff;
222	if(ix>=0x7ff00000) return x*x;
223	if((ix|lx)==0) return one/zero;
224	if(ix<0x3b900000) {	/* |x|<2**-70, return -ieee_log(|x|) */
225	    if(hx<0) {
226	        *signgamp = -1;
227	        return -__ieee754_log(-x);
228	    } else return -__ieee754_log(x);
229	}
230	if(hx<0) {
231	    if(ix>=0x43300000) 	/* |x|>=2**52, must be -integer */
232		return one/zero;
233	    t = sin_pi(x);
234	    if(t==zero) return one/zero; /* -integer */
235	    nadj = __ieee754_log(pi/ieee_fabs(t*x));
236	    if(t<zero) *signgamp = -1;
237	    x = -x;
238	}
239
240    /* purge off 1 and 2 */
241	if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
242    /* for x < 2.0 */
243	else if(ix<0x40000000) {
244	    if(ix<=0x3feccccc) { 	/* ieee_lgamma(x) = ieee_lgamma(x+1)-ieee_log(x) */
245		r = -__ieee754_log(x);
246		if(ix>=0x3FE76944) {y = one-x; i= 0;}
247		else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
248	  	else {y = x; i=2;}
249	    } else {
250	  	r = zero;
251	        if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
252	        else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
253		else {y=x-one;i=2;}
254	    }
255	    switch(i) {
256	      case 0:
257		z = y*y;
258		p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
259		p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
260		p  = y*p1+p2;
261		r  += (p-0.5*y); break;
262	      case 1:
263		z = y*y;
264		w = z*y;
265		p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));	/* parallel comp */
266		p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
267		p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
268		p  = z*p1-(tt-w*(p2+y*p3));
269		r += (tf + p); break;
270	      case 2:
271		p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
272		p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
273		r += (-0.5*y + p1/p2);
274	    }
275	}
276	else if(ix<0x40200000) { 			/* x < 8.0 */
277	    i = (int)x;
278	    t = zero;
279	    y = x-(double)i;
280	    p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
281	    q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
282	    r = half*y+p/q;
283	    z = one;	/* ieee_lgamma(1+s) = ieee_log(s) + ieee_lgamma(s) */
284	    switch(i) {
285	    case 7: z *= (y+6.0);	/* FALLTHRU */
286	    case 6: z *= (y+5.0);	/* FALLTHRU */
287	    case 5: z *= (y+4.0);	/* FALLTHRU */
288	    case 4: z *= (y+3.0);	/* FALLTHRU */
289	    case 3: z *= (y+2.0);	/* FALLTHRU */
290		    r += __ieee754_log(z); break;
291	    }
292    /* 8.0 <= x < 2**58 */
293	} else if (ix < 0x43900000) {
294	    t = __ieee754_log(x);
295	    z = one/x;
296	    y = z*z;
297	    w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
298	    r = (x-half)*(t-one)+w;
299	} else
300    /* 2**58 <= x <= inf */
301	    r =  x*(__ieee754_log(x)-one);
302	if(hx<0) r = nadj - r;
303	return r;
304}
305