2 /* @(#)e_lgamma_r.c 1.3 95/01/18 */
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
11 * ====================================================
16 static char rcsid
[] = "$FreeBSD: src/lib/msun/src/e_lgamma_r.c,v 1.9 2007/05/02 16:54:22 bde Exp $";
19 /* __ieee754_lgamma_r(x, signgamp)
20 * Reentrant version of the logarithm of the Gamma function
21 * with user provide pointer for the sign of Gamma(x).
24 * 1. Argument Reduction for 0 < x <= 8
25 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
26 * reduce x to a number in [1.5,2.5] by
27 * lgamma(1+s) = log(s) + lgamma(s)
29 * lgamma(7.3) = log(6.3) + lgamma(6.3)
30 * = log(6.3*5.3) + lgamma(5.3)
31 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
32 * 2. Polynomial approximation of lgamma around its
33 * minimun ymin=1.461632144968362245 to maintain monotonicity.
34 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
36 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
38 * poly(z) is a 14 degree polynomial.
39 * 2. Rational approximation in the primary interval [2,3]
40 * We use the following approximation:
42 * lgamma(x) = 0.5*s + s*P(s)/Q(s)
44 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
45 * Our algorithms are based on the following observation
47 * zeta(2)-1 2 zeta(3)-1 3
48 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
51 * where Euler = 0.5771... is the Euler constant, which is very
54 * 3. For x>=8, we have
55 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
57 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
58 * Let z = 1/x, then we approximation
59 * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
62 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
64 * |w - f(z)| < 2**-58.74
66 * 4. For negative x, since (G is gamma function)
67 * -x*G(-x)*G(x) = pi/sin(pi*x),
69 * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
70 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
71 * Hence, for x<0, signgam = sign(sin(pi*x)) and
72 * lgamma(x) = log(|Gamma(x)|)
73 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
74 * Note: one should avoid compute pi*(-x) directly in the
75 * computation of sin(pi*(-x)).
78 * lgamma(2+s) ~ s*(1-Euler) for tiny s
79 * lgamma(1) = lgamma(2) = 0
80 * lgamma(x) ~ -log(|x|) for tiny x
81 * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
83 * lgamma(-inf) = inf (bug for bug compatible with C99!?)
88 #include "math_private.h"
91 two52
= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
92 half
= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
93 one
= 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
94 pi
= 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
95 a0
= 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
96 a1
= 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
97 a2
= 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
98 a3
= 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
99 a4
= 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
100 a5
= 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
101 a6
= 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
102 a7
= 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
103 a8
= 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
104 a9
= 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
105 a10
= 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
106 a11
= 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
107 tc
= 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
108 tf
= -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
109 /* tt = -(tail of tf) */
110 tt
= -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
111 t0
= 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
112 t1
= -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
113 t2
= 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
114 t3
= -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
115 t4
= 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
116 t5
= -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
117 t6
= 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
118 t7
= -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
119 t8
= 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
120 t9
= -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
121 t10
= 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
122 t11
= -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
123 t12
= 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
124 t13
= -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
125 t14
= 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
126 u0
= -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
127 u1
= 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
128 u2
= 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
129 u3
= 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
130 u4
= 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
131 u5
= 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
132 v1
= 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
133 v2
= 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
134 v3
= 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
135 v4
= 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
136 v5
= 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
137 s0
= -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
138 s1
= 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
139 s2
= 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
140 s3
= 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
141 s4
= 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
142 s5
= 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
143 s6
= 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
144 r1
= 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
145 r2
= 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
146 r3
= 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
147 r4
= 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
148 r5
= 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
149 r6
= 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
150 w0
= 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
151 w1
= 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
152 w2
= -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
153 w3
= 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
154 w4
= -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
155 w5
= 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
156 w6
= -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
158 static const double zero
= 0.00000000000000000000e+00;
160 static double sin_pi(double x
)
168 if(ix
<0x3fd00000) return __kernel_sin(pi
*x
,zero
,0);
169 y
= -x
; /* x is assume negative */
172 * argument reduction, make sure inexact flag not raised if input
176 if(z
!=y
) { /* inexact anyway */
178 y
= 2.0*(y
- floor(y
)); /* y = |x| mod 2.0 */
182 y
= zero
; n
= 0; /* y must be even */
184 if(ix
<0x43300000) z
= y
+two52
; /* exact */
192 case 0: y
= __kernel_sin(pi
*y
,zero
,0); break;
194 case 2: y
= __kernel_cos(pi
*(0.5-y
),zero
); break;
196 case 4: y
= __kernel_sin(pi
*(one
-y
),zero
,0); break;
198 case 6: y
= -__kernel_cos(pi
*(y
-1.5),zero
); break;
199 default: y
= __kernel_sin(pi
*(y
-2.0),zero
,0); break;
206 __ieee754_lgamma_r(double x
, int *signgamp
)
208 double t
,y
,z
,nadj
,p
,p1
,p2
,p3
,q
,r
,w
;
212 EXTRACT_WORDS(hx
,lx
,x
);
214 /* purge off +-inf, NaN, +-0, tiny and negative arguments */
217 if(ix
>=0x7ff00000) return x
*x
;
218 if((ix
|lx
)==0) return one
/zero
;
219 if(ix
<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
222 return -__ieee754_log(-x
);
223 } else return -__ieee754_log(x
);
226 if(ix
>=0x43300000) /* |x|>=2**52, must be -integer */
229 if(t
==zero
) return one
/zero
; /* -integer */
230 nadj
= __ieee754_log(pi
/fabs(t
*x
));
231 if(t
<zero
) *signgamp
= -1;
235 /* purge off 1 and 2 */
236 if((((ix
-0x3ff00000)|lx
)==0)||(((ix
-0x40000000)|lx
)==0)) r
= 0;
238 else if(ix
<0x40000000) {
239 if(ix
<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
240 r
= -__ieee754_log(x
);
241 if(ix
>=0x3FE76944) {y
= one
-x
; i
= 0;}
242 else if(ix
>=0x3FCDA661) {y
= x
-(tc
-one
); i
=1;}
246 if(ix
>=0x3FFBB4C3) {y
=2.0-x
;i
=0;} /* [1.7316,2] */
247 else if(ix
>=0x3FF3B4C4) {y
=x
-tc
;i
=1;} /* [1.23,1.73] */
253 p1
= a0
+z
*(a2
+z
*(a4
+z
*(a6
+z
*(a8
+z
*a10
))));
254 p2
= z
*(a1
+z
*(a3
+z
*(a5
+z
*(a7
+z
*(a9
+z
*a11
)))));
256 r
+= (p
-0.5*y
); break;
260 p1
= t0
+w
*(t3
+w
*(t6
+w
*(t9
+w
*t12
))); /* parallel comp */
261 p2
= t1
+w
*(t4
+w
*(t7
+w
*(t10
+w
*t13
)));
262 p3
= t2
+w
*(t5
+w
*(t8
+w
*(t11
+w
*t14
)));
263 p
= z
*p1
-(tt
-w
*(p2
+y
*p3
));
264 r
+= (tf
+ p
); break;
266 p1
= y
*(u0
+y
*(u1
+y
*(u2
+y
*(u3
+y
*(u4
+y
*u5
)))));
267 p2
= one
+y
*(v1
+y
*(v2
+y
*(v3
+y
*(v4
+y
*v5
))));
268 r
+= (-0.5*y
+ p1
/p2
);
271 else if(ix
<0x40200000) { /* x < 8.0 */
275 p
= y
*(s0
+y
*(s1
+y
*(s2
+y
*(s3
+y
*(s4
+y
*(s5
+y
*s6
))))));
276 q
= one
+y
*(r1
+y
*(r2
+y
*(r3
+y
*(r4
+y
*(r5
+y
*r6
)))));
278 z
= one
; /* lgamma(1+s) = log(s) + lgamma(s) */
280 case 7: z
*= (y
+6.0); /* FALLTHRU */
281 case 6: z
*= (y
+5.0); /* FALLTHRU */
282 case 5: z
*= (y
+4.0); /* FALLTHRU */
283 case 4: z
*= (y
+3.0); /* FALLTHRU */
284 case 3: z
*= (y
+2.0); /* FALLTHRU */
285 r
+= __ieee754_log(z
); break;
287 /* 8.0 <= x < 2**58 */
288 } else if (ix
< 0x43900000) {
289 t
= __ieee754_log(x
);
292 w
= w0
+z
*(w1
+y
*(w2
+y
*(w3
+y
*(w4
+y
*(w5
+y
*w6
)))));
293 r
= (x
-half
)*(t
-one
)+w
;
295 /* 2**58 <= x <= inf */
296 r
= x
*(__ieee754_log(x
)-one
);
297 if(hx
<0) r
= nadj
- r
;