e_lgamma_r.c revision 97413 
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
14static char rcsid[] = "$FreeBSD: head/lib/msun/src/e_lgamma_r.c 97413 2002-05-28 18:15:04Z alfred $";
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
87static const double
88two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
89half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
90one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
91pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
92a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
93a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
94a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
95a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
96a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
97a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
98a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
99a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
100a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
101a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
102a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
103a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
104tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
105tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
106/* tt = -(tail of tf) */
107tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
108t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
109t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
110t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
111t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
112t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
113t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
114t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
115t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
116t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
117t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
118t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
119t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
120t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
121t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
122t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
123u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
124u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
125u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
126u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
127u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
128u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
129v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
130v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
131v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
132v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
133v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
134s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
135s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
136s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
137s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
138s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
139s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
140s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
141r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
142r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
143r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
144r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
145r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
146r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
147w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
148w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
149w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
150w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
151w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
152w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
153w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
154
155static 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
202double
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}
296