e_lgamma_r.c revision 271719 
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#include <sys/cdefs.h>
16__FBSDID("$FreeBSD: head/lib/msun/src/e_lgamma_r.c 271719 2014-09-17 19:01:22Z kargl $");
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(neg.integer) = inf and raise divide-by-zero
81 *		lgamma(inf) = inf
82 *		lgamma(-inf) = inf (bug for bug compatible with C99!?)
83 *
84 */
85
86#include <float.h>
87
88#include "math.h"
89#include "math_private.h"
90
91static const volatile double vzero = 0;
92
93static const double
94zero=  0.00000000000000000000e+00,
95half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
96one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
97pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
98a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
99a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
100a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
101a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
102a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
103a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
104a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
105a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
106a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
107a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
108a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
109a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
110tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
111tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
112/* tt = -(tail of tf) */
113tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
114t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
115t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
116t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
117t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
118t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
119t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
120t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
121t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
122t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
123t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
124t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
125t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
126t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
127t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
128t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
129u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
130u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
131u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
132u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
133u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
134u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
135v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
136v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
137v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
138v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
139v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
140s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
141s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
142s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
143s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
144s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
145s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
146s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
147r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
148r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
149r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
150r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
151r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
152r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
153w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
154w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
155w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
156w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
157w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
158w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
159w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
160
161/*
162 * Compute sin(pi*x) without actually doing the pi*x multiplication.
163 * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is
164 * the precision of x.
165 */
166static double
167sin_pi(double x)
168{
169	volatile double vz;
170	double y,z;
171	int n;
172
173	y = -x;
174
175	vz = y+0x1p52;			/* depend on 0 <= y < 0x1p52 */
176	z = vz-0x1p52;			/* rint(y) for the above range */
177	if (z == y)
178	    return zero;
179
180	vz = y+0x1p50;
181	GET_LOW_WORD(n,vz);		/* bits for rounded y (units 0.25) */
182	z = vz-0x1p50;			/* y rounded to a multiple of 0.25 */
183	if (z > y) {
184	    z -= 0.25;			/* adjust to round down */
185	    n--;
186	}
187	n &= 7;				/* octant of y mod 2 */
188	y = y - z + n * 0.25;		/* y mod 2 */
189
190	switch (n) {
191	    case 0:   y =  __kernel_sin(pi*y,zero,0); break;
192	    case 1:
193	    case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
194	    case 3:
195	    case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
196	    case 5:
197	    case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
198	    default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
199	    }
200	return -y;
201}
202
203
204double
205__ieee754_lgamma_r(double x, int *signgamp)
206{
207	double t,y,z,nadj,p,p1,p2,p3,q,r,w;
208	int32_t hx;
209	int i,ix,lx;
210
211	EXTRACT_WORDS(hx,lx,x);
212
213    /* purge off +-inf, NaN, +-0, tiny and negative arguments */
214	*signgamp = 1;
215	ix = hx&0x7fffffff;
216	if(ix>=0x7ff00000) return x*x;
217	if((ix|lx)==0) {
218	   if(hx<0)
219		*signgamp = -1;
220	    return one/vzero;
221	}
222	if(ix<0x3b900000) {	/* |x|<2**-70, return -log(|x|) */
223	    if(hx<0) {
224	        *signgamp = -1;
225	        return -__ieee754_log(-x);
226	    } else return -__ieee754_log(x);
227	}
228	if(hx<0) {
229	    if(ix>=0x43300000) 	/* |x|>=2**52, must be -integer */
230		return one/vzero;
231	    t = sin_pi(x);
232	    if(t==zero) return one/vzero; /* -integer */
233	    nadj = __ieee754_log(pi/fabs(t*x));
234	    if(t<zero) *signgamp = -1;
235	    x = -x;
236	}
237
238    /* purge off 1 and 2 */
239	if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
240    /* for x < 2.0 */
241	else if(ix<0x40000000) {
242	    if(ix<=0x3feccccc) { 	/* lgamma(x) = lgamma(x+1)-log(x) */
243		r = -__ieee754_log(x);
244		if(ix>=0x3FE76944) {y = one-x; i= 0;}
245		else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
246	  	else {y = x; i=2;}
247	    } else {
248	  	r = zero;
249	        if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
250	        else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
251		else {y=x-one;i=2;}
252	    }
253	    switch(i) {
254	      case 0:
255		z = y*y;
256		p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
257		p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
258		p  = y*p1+p2;
259		r  += (p-y/2); break;
260	      case 1:
261		z = y*y;
262		w = z*y;
263		p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));	/* parallel comp */
264		p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
265		p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
266		p  = z*p1-(tt-w*(p2+y*p3));
267		r += (tf + p); break;
268	      case 2:
269		p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
270		p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
271		r += (-0.5*y + p1/p2);
272	    }
273	}
274	else if(ix<0x40200000) { 			/* x < 8.0 */
275	    i = (int)x;
276	    y = x-(double)i;
277	    p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
278	    q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
279	    r = half*y+p/q;
280	    z = one;	/* lgamma(1+s) = log(s) + lgamma(s) */
281	    switch(i) {
282	    case 7: z *= (y+6);		/* FALLTHRU */
283	    case 6: z *= (y+5);		/* FALLTHRU */
284	    case 5: z *= (y+4);		/* FALLTHRU */
285	    case 4: z *= (y+3);		/* FALLTHRU */
286	    case 3: z *= (y+2);		/* FALLTHRU */
287		    r += __ieee754_log(z); break;
288	    }
289    /* 8.0 <= x < 2**58 */
290	} else if (ix < 0x43900000) {
291	    t = __ieee754_log(x);
292	    z = one/x;
293	    y = z*z;
294	    w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
295	    r = (x-half)*(t-one)+w;
296	} else
297    /* 2**58 <= x <= inf */
298	    r =  x*(__ieee754_log(x)-one);
299	if(hx<0) r = nadj - r;
300	return r;
301}
302
303#if (LDBL_MANT_DIG == 53)
304__weak_reference(lgamma_r, lgammal_r);
305#endif
306
307