GCC Code Coverage Report
Directory: ./ Exec Total Coverage
File: lib/libm/src/e_lgamma_r.c Lines: 88 97 90.7 %
Date: 2017-11-13 Branches: 55 66 83.3 %

Line Branch Exec Source
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/* @(#)er_lgamma.c 5.1 93/09/24 */
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/*
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 * ====================================================
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 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5
 *
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 * Developed at SunPro, a Sun Microsystems, Inc. business.
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 * Permission to use, copy, modify, and distribute this
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 * software is freely granted, provided that this notice
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 * is preserved.
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 * ====================================================
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 */
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/* lgamma_r(x, signgamp)
14
 * Reentrant version of the logarithm of the Gamma function
15
 * with user provide pointer for the sign of Gamma(x).
16
 *
17
 * Method:
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 *   1. Argument Reduction for 0 < x <= 8
19
 * 	Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
20
 * 	reduce x to a number in [1.5,2.5] by
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 * 		lgamma(1+s) = log(s) + lgamma(s)
22
 *	for example,
23
 *		lgamma(7.3) = log(6.3) + lgamma(6.3)
24
 *			    = log(6.3*5.3) + lgamma(5.3)
25
 *			    = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
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 *   2. Polynomial approximation of lgamma around its
27
 *	minimun ymin=1.461632144968362245 to maintain monotonicity.
28
 *	On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
29
 *		Let z = x-ymin;
30
 *		lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
31
 *	where
32
 *		poly(z) is a 14 degree polynomial.
33
 *   2. Rational approximation in the primary interval [2,3]
34
 *	We use the following approximation:
35
 *		s = x-2.0;
36
 *		lgamma(x) = 0.5*s + s*P(s)/Q(s)
37
 *	with accuracy
38
 *		|P/Q - (lgamma(x)-0.5s)| < 2**-61.71
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 *	Our algorithms are based on the following observation
40
 *
41
 *                             zeta(2)-1    2    zeta(3)-1    3
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 * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
43
 *                                 2                 3
44
 *
45
 *	where Euler = 0.5771... is the Euler constant, which is very
46
 *	close to 0.5.
47
 *
48
 *   3. For x>=8, we have
49
 *	lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
50
 *	(better formula:
51
 *	   lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
52
 *	Let z = 1/x, then we approximation
53
 *		f(z) = lgamma(x) - (x-0.5)(log(x)-1)
54
 *	by
55
 *	  			    3       5             11
56
 *		w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
57
 *	where
58
 *		|w - f(z)| < 2**-58.74
59
 *
60
 *   4. For negative x, since (G is gamma function)
61
 *		-x*G(-x)*G(x) = pi/sin(pi*x),
62
 * 	we have
63
 * 		G(x) = pi/(sin(pi*x)*(-x)*G(-x))
64
 *	since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
65
 *	Hence, for x<0, signgam = sign(sin(pi*x)) and
66
 *		lgamma(x) = log(|Gamma(x)|)
67
 *			  = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
68
 *	Note: one should avoid compute pi*(-x) directly in the
69
 *	      computation of sin(pi*(-x)).
70
 *
71
 *   5. Special Cases
72
 *		lgamma(2+s) ~ s*(1-Euler) for tiny s
73
 *		lgamma(1)=lgamma(2)=0
74
 *		lgamma(x) ~ -log(x) for tiny x
75
 *		lgamma(0) = lgamma(inf) = inf
76
 *	 	lgamma(-integer) = +-inf
77
 *
78
 */
79
80
#include "math.h"
81
#include "math_private.h"
82
83
static const double
84
two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
85
half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
86
one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
87
pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
88
a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
89
a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
90
a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
91
a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
92
a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
93
a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
94
a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
95
a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
96
a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
97
a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
98
a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
99
a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
100
tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
101
tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
102
/* tt = -(tail of tf) */
103
tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
104
t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
105
t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
106
t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
107
t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
108
t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
109
t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
110
t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
111
t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
112
t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
113
t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
114
t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
115
t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
116
t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
117
t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
118
t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
119
u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
120
u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
121
u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
122
u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
123
u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
124
u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
125
v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
126
v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
127
v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
128
v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
129
v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
130
s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
131
s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
132
s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
133
s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
134
s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
135
s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
136
s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
137
r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
138
r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
139
r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
140
r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
141
r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
142
r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
143
w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
144
w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
145
w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
146
w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
147
w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
148
w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
149
w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
150
151
static const double zero=  0.00000000000000000000e+00;
152
153
static double
154
sin_pi(double x)
155
{
156
	double y,z;
157
	int n,ix;
158
159
6948
	GET_HIGH_WORD(ix,x);
160
3474
	ix &= 0x7fffffff;
161
162
4524
	if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
163
2424
	y = -x;		/* x is assume negative */
164
165
    /*
166
     * argument reduction, make sure inexact flag not raised if input
167
     * is an integer
168
     */
169
2424
	z = floor(y);
170
2424
	if(z!=y) {				/* inexact anyway */
171
2424
	    y  *= 0.5;
172
2424
	    y   = 2.0*(y - floor(y));		/* y = |x| mod 2.0 */
173
2424
	    n   = (int) (y*4.0);
174
2424
	} else {
175
            if(ix>=0x43400000) {
176
                y = zero; n = 0;                 /* y must be even */
177
            } else {
178
                if(ix<0x43300000) z = y+two52;	/* exact */
179
		GET_LOW_WORD(n,z);
180
		n &= 1;
181
                y  = n;
182
                n<<= 2;
183
            }
184
        }
185


2424
	switch (n) {
186
78
	    case 0:   y =  __kernel_sin(pi*y,zero,0); break;
187
	    case 1:
188
276
	    case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
189
	    case 3:
190
972
	    case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
191
	    case 5:
192
630
	    case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
193
468
	    default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
194
	    }
195
2424
	return -y;
196
3474
}
197
198
199
double
200
lgamma_r(double x, int *signgamp)
201
{
202
	double t,y,z,nadj,p,p1,p2,p3,q,r,w;
203
	int i,hx,lx,ix;
204
205
35838
	EXTRACT_WORDS(hx,lx,x);
206
207
    /* purge off +-inf, NaN, +-0, and negative arguments */
208
17919
	*signgamp = 1;
209
17919
	ix = hx&0x7fffffff;
210
17928
	if(ix>=0x7ff00000) return x*x;
211
17910
	if((ix|lx)==0) {
212
6
	    if(hx<0)
213
3
		*signgamp = -1;
214
6
	    return one/zero;
215
	}
216
17904
	if(ix<0x3b900000) {	/* |x|<2**-70, return -log(|x|) */
217
48
	    if(hx<0) {
218
24
	        *signgamp = -1;
219
24
	        return - log(-x);
220
24
	    } else return - log(x);
221
	}
222
17856
	if(hx<0) {
223
3474
	    if(ix>=0x43300000) 	/* |x|>=2**52, must be -integer */
224
		return one/zero;
225
3474
	    t = sin_pi(x);
226
3474
	    if(t==zero) return one/zero; /* -integer */
227
3474
	    nadj = log(pi/fabs(t*x));
228
5166
	    if(t<zero) *signgamp = -1;
229
3474
	    x = -x;
230
3474
	}
231
232
    /* purge off 1 and 2 */
233

35718
	if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
234
    /* for x < 2.0 */
235
17841
	else if(ix<0x40000000) {
236
11094
	    if(ix<=0x3feccccc) { 	/* lgamma(x) = lgamma(x+1)-log(x) */
237
4284
		r = - log(x);
238
4548
		if(ix>=0x3FE76944) {y = one-x; i= 0;}
239
5412
		else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
240
	  	else {y = x; i=2;}
241
	    } else {
242
	  	r = zero;
243
8568
	        if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
244
6588
	        else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
245
3516
		else {y=x-one;i=2;}
246
	    }
247

22200
	    switch(i) {
248
	      case 0:
249
2022
		z = y*y;
250
2022
		p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
251
2022
		p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
252
2022
		p  = y*p1+p2;
253
2022
		r  += (p-0.5*y); break;
254
	      case 1:
255
2928
		z = y*y;
256
2928
		w = z*y;
257
2928
		p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));	/* parallel comp */
258
2928
		p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
259
2928
		p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
260
2928
		p  = z*p1-(tt-w*(p2+y*p3));
261
2928
		r += (tf + p); break;
262
	      case 2:
263
6144
		p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
264
6144
		p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
265
6144
		r += (-0.5*y + p1/p2);
266
6144
	    }
267
	}
268
6747
	else if(ix<0x40200000) { 			/* x < 8.0 */
269
9906
	    i = (int)x;
270
	    t = zero;
271
9906
	    y = x-(double)i;
272
9906
	    p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
273
9906
	    q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
274
9906
	    r = half*y+p/q;
275
	    z = one;	/* lgamma(1+s) = log(s) + lgamma(s) */
276

9906
	    switch(i) {
277
192
	    case 7: z *= (y+6.0);	/* FALLTHRU */
278
258
	    case 6: z *= (y+5.0);	/* FALLTHRU */
279
414
	    case 5: z *= (y+4.0);	/* FALLTHRU */
280
582
	    case 4: z *= (y+3.0);	/* FALLTHRU */
281
1713
	    case 3: z *= (y+2.0);	/* FALLTHRU */
282
1713
		    r += log(z); break;
283
	    }
284
    /* 8.0 <= x < 2**58 */
285
3450
	} else if (ix < 0x43900000) {
286
3450
	    t = log(x);
287
3450
	    z = one/x;
288
3450
	    y = z*z;
289
3450
	    w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
290
3450
	    r = (x-half)*(t-one)+w;
291
3450
	} else
292
    /* 2**58 <= x <= inf */
293
	    r =  x*(log(x)-one);
294
21330
	if(hx<0) r = nadj - r;
295
17856
	return r;
296
17919
}
297
DEF_NONSTD(lgamma_r);