1 //===========================================================================
5 // Part of the standard mathematical function library
7 //===========================================================================
8 //####ECOSGPLCOPYRIGHTBEGIN####
9 // -------------------------------------------
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38 // -------------------------------------------
39 //####ECOSGPLCOPYRIGHTEND####
40 //===========================================================================
41 //#####DESCRIPTIONBEGIN####
43 // Author(s): jlarmour
44 // Contributors: jlarmour
50 //####DESCRIPTIONEND####
52 //===========================================================================
56 #include <pkgconf/libm.h> // Configuration header
58 // Include the Math library?
61 // Derived from code with the following copyright
64 /* @(#)e_lgamma_r.c 1.3 95/01/18 */
66 * ====================================================
67 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
69 * Developed at SunSoft, a Sun Microsystems, Inc. business.
70 * Permission to use, copy, modify, and distribute this
71 * software is freely granted, provided that this notice
73 * ====================================================
77 /* __ieee754_lgamma_r(x, signgamp)
78 * Reentrant version of the logarithm of the Gamma function
79 * with user provide pointer for the sign of Gamma(x).
82 * 1. Argument Reduction for 0 < x <= 8
83 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
84 * reduce x to a number in [1.5,2.5] by
85 * lgamma(1+s) = log(s) + lgamma(s)
87 * lgamma(7.3) = log(6.3) + lgamma(6.3)
88 * = log(6.3*5.3) + lgamma(5.3)
89 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
90 * 2. Polynomial approximation of lgamma around its
91 * minimun ymin=1.461632144968362245 to maintain monotonicity.
92 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
94 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
96 * poly(z) is a 14 degree polynomial.
97 * 2. Rational approximation in the primary interval [2,3]
98 * We use the following approximation:
100 * lgamma(x) = 0.5*s + s*P(s)/Q(s)
102 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
103 * Our algorithms are based on the following observation
105 * zeta(2)-1 2 zeta(3)-1 3
106 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
109 * where Euler = 0.5771... is the Euler constant, which is very
112 * 3. For x>=8, we have
113 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
115 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
116 * Let z = 1/x, then we approximation
117 * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
120 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
122 * |w - f(z)| < 2**-58.74
124 * 4. For negative x, since (G is gamma function)
125 * -x*G(-x)*G(x) = pi/sin(pi*x),
127 * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
128 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
129 * Hence, for x<0, signgam = sign(sin(pi*x)) and
130 * lgamma(x) = log(|Gamma(x)|)
131 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
132 * Note: one should avoid compute pi*(-x) directly in the
133 * computation of sin(pi*(-x)).
136 * lgamma(2+s) ~ s*(1-Euler) for tiny s
137 * lgamma(1)=lgamma(2)=0
138 * lgamma(x) ~ -log(x) for tiny x
139 * lgamma(0) = lgamma(inf) = inf
140 * lgamma(-integer) = +-inf
144 #include "mathincl/fdlibm.h"
147 two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
148 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
149 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
150 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
151 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
152 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
153 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
154 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
155 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
156 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
157 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
158 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
159 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
160 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
161 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
162 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
163 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
164 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
165 /* tt = -(tail of tf) */
166 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
167 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
168 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
169 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
170 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
171 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
172 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
173 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
174 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
175 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
176 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
177 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
178 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
179 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
180 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
181 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
182 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
183 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
184 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
185 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
186 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
187 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
188 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
189 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
190 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
191 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
192 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
193 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
194 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
195 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
196 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
197 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
198 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
199 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
200 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
201 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
202 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
203 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
204 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
205 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
206 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
207 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
208 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
209 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
210 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
211 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
212 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
214 static double zero= 0.00000000000000000000e+00;
216 static double sin_pi(double x)
221 ix = 0x7fffffff&CYG_LIBM_HI(x);
223 if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
224 y = -x; /* x is assume negative */
227 * argument reduction, make sure inexact flag not raised if input
231 if(z!=y) { /* inexact anyway */
233 y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */
237 y = zero; n = 0; /* y must be even */
239 if(ix<0x43300000) z = y+two52; /* exact */
240 n = CYG_LIBM_LO(z)&1; /* lower word of z */
246 case 0: y = __kernel_sin(pi*y,zero,0); break;
248 case 2: y = __kernel_cos(pi*(0.5-y),zero); break;
250 case 4: y = __kernel_sin(pi*(one-y),zero,0); break;
252 case 6: y = -__kernel_cos(pi*(y-1.5),zero); break;
253 default: y = __kernel_sin(pi*(y-2.0),zero,0); break;
259 double __ieee754_lgamma_r(double x, int *signgamp)
261 double t,y,z,nadj,p,p1,p2,p3,q,r,w;
264 nadj = 0.0; /* to placate compiler */
268 /* purge off +-inf, NaN, +-0, and negative arguments */
271 if(ix>=0x7ff00000) return x*x;
272 if((ix|lx)==0) return one/zero;
273 if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
276 return -__ieee754_log(-x);
277 } else return -__ieee754_log(x);
280 if(ix>=0x43300000) /* |x|>=2**52, must be -integer */
283 if(t==zero) return one/zero; /* -integer */
284 nadj = __ieee754_log(pi/fabs(t*x));
285 if(t<zero) *signgamp = -1;
289 /* purge off 1 and 2 */
290 if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
292 else if(ix<0x40000000) {
293 if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
294 r = -__ieee754_log(x);
295 if(ix>=0x3FE76944) {y = one-x; i= 0;}
296 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
300 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
301 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
307 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
308 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
310 r += (p-0.5*y); break;
314 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
315 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
316 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
317 p = z*p1-(tt-w*(p2+y*p3));
318 r += (tf + p); break;
320 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
321 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
322 r += (-0.5*y + p1/p2);
325 else if(ix<0x40200000) { /* x < 8.0 */
329 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
330 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
332 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
334 case 7: z *= (y+6.0); /* FALLTHRU */
335 case 6: z *= (y+5.0); /* FALLTHRU */
336 case 5: z *= (y+4.0); /* FALLTHRU */
337 case 4: z *= (y+3.0); /* FALLTHRU */
338 case 3: z *= (y+2.0); /* FALLTHRU */
339 r += __ieee754_log(z); break;
341 /* 8.0 <= x < 2**58 */
342 } else if (ix < 0x43900000) {
343 t = __ieee754_log(x);
346 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
347 r = (x-half)*(t-one)+w;
349 /* 2**58 <= x <= inf */
350 r = x*(__ieee754_log(x)-one);
351 if(hx<0) r = nadj - r;
355 #endif // ifdef CYGPKG_LIBM