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1a3b704c 1/* @(#)e_lgamma_r.c 1.3 95/01/18 */
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2/*
3 * ====================================================
5 *
1a3b704c 6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
b34b60bc 7 * Permission to use, copy, modify, and distribute this
2fedfd5c 8 * software is freely granted, provided that this notice
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9 * is preserved.
10 * ====================================================
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11 */
12
2fedfd5c 13
6ff43c94 14/* __ieee754_lgamma_r(x, signgamp)
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15 * Reentrant version of the logarithm of the Gamma function
16 * with user provide pointer for the sign of Gamma(x).
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17 *
18 * Method:
19 * 1. Argument Reduction for 0 < x <= 8
2fedfd5c 20 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
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21 * reduce x to a number in [1.5,2.5] by
22 * lgamma(1+s) = log(s) + lgamma(s)
23 * for example,
24 * lgamma(7.3) = log(6.3) + lgamma(6.3)
25 * = log(6.3*5.3) + lgamma(5.3)
26 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
27 * 2. Polynomial approximation of lgamma around its
28 * minimun ymin=1.461632144968362245 to maintain monotonicity.
29 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
30 * Let z = x-ymin;
31 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
32 * where
33 * poly(z) is a 14 degree polynomial.
34 * 2. Rational approximation in the primary interval [2,3]
35 * We use the following approximation:
36 * s = x-2.0;
37 * lgamma(x) = 0.5*s + s*P(s)/Q(s)
38 * with accuracy
39 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
40 * Our algorithms are based on the following observation
41 *
42 * zeta(2)-1 2 zeta(3)-1 3
43 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
44 * 2 3
45 *
46 * where Euler = 0.5771... is the Euler constant, which is very
47 * close to 0.5.
48 *
49 * 3. For x>=8, we have
50 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
51 * (better formula:
52 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
53 * Let z = 1/x, then we approximation
54 * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
55 * by
56 * 3 5 11
57 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
2fedfd5c 58 * where
b34b60bc 59 * |w - f(z)| < 2**-58.74
2fedfd5c 60 *
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61 * 4. For negative x, since (G is gamma function)
62 * -x*G(-x)*G(x) = pi/sin(pi*x),
63 * we have
64 * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
65 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
2fedfd5c 66 * Hence, for x<0, signgam = sign(sin(pi*x)) and
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67 * lgamma(x) = log(|Gamma(x)|)
68 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
2fedfd5c 69 * Note: one should avoid compute pi*(-x) directly in the
b34b60bc 70 * computation of sin(pi*(-x)).
2fedfd5c 71 *
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72 * 5. Special Cases
73 * lgamma(2+s) ~ s*(1-Euler) for tiny s
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74 * lgamma(1) = lgamma(2) = 0
75 * lgamma(x) ~ -log(|x|) for tiny x
76 * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
77 * lgamma(inf) = inf
78 * lgamma(-inf) = inf (bug for bug compatible with C99!?)
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79 */
80
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81#include <float.h>
82
6ff43c94 83#include "math.h"
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84#include "math_private.h"
85
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86static const volatile double vzero = 0;
87
b34b60bc 88static const double
2fedfd5c 89zero= 0.00000000000000000000e+00,
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90half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
91one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
92pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
93a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
94a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
95a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
96a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
97a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
98a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
99a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
100a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
101a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
102a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
103a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
104a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
105tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
106tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
107/* tt = -(tail of tf) */
108tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
109t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
110t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
111t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
112t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
113t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
114t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
115t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
116t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
117t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
118t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
119t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
120t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
121t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
122t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
123t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
124u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
125u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
126u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
127u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
128u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
129u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
130v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
131v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
132v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
133v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
134v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
135s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
136s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
137s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
138s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
139s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
140s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
141s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
142r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
143r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
144r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
145r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
146r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
147r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
148w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
149w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
150w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
151w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
152w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
153w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
154w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
155
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156/*
157 * Compute sin(pi*x) without actually doing the pi*x multiplication.
158 * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is
159 * the precision of x.
160 */
161static double
162sin_pi(double x)
b34b60bc 163{
2fedfd5c 164 volatile double vz;
b34b60bc 165 double y,z;
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166 int n;
167
168 y = -x;
b34b60bc 169
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170 vz = y+0x1p52; /* depend on 0 <= y < 0x1p52 */
171 z = vz-0x1p52; /* rint(y) for the above range */
172 if (z == y)
173 return zero;
b34b60bc 174
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175 vz = y+0x1p50;
176 GET_LOW_WORD(n,vz); /* bits for rounded y (units 0.25) */
177 z = vz-0x1p50; /* y rounded to a multiple of 0.25 */
178 if (z > y) {
179 z -= 0.25; /* adjust to round down */
180 n--;
181 }
182 n &= 7; /* octant of y mod 2 */
183 y = y - z + n * 0.25; /* y mod 2 */
b34b60bc 184
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185 switch (n) {
186 case 0: y = __kernel_sin(pi*y,zero,0); break;
2fedfd5c 187 case 1:
b34b60bc 188 case 2: y = __kernel_cos(pi*(0.5-y),zero); break;
2fedfd5c 189 case 3:
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190 case 4: y = __kernel_sin(pi*(one-y),zero,0); break;
191 case 5:
192 case 6: y = -__kernel_cos(pi*(y-1.5),zero); break;
193 default: y = __kernel_sin(pi*(y-2.0),zero,0); break;
194 }
195 return -y;
196}
197
198
199double
6ff43c94 200__ieee754_lgamma_r(double x, int *signgamp)
b34b60bc 201{
1a3b704c 203 int32_t hx;
2fedfd5c 204 int i,ix,lx;
b34b60bc 205
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206 EXTRACT_WORDS(hx,lx,x);
207
2fedfd5c 208 /* purge +-Inf and NaNs */
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209 *signgamp = 1;
210 ix = hx&0x7fffffff;
211 if(ix>=0x7ff00000) return x*x;
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212
213 /* purge +-0 and tiny arguments */
214 *signgamp = 1-2*((uint32_t)hx>>31);
215 if(ix<0x3c700000) { /* |x|<2**-56, return -log(|x|) */
216 if((ix|lx)==0)
217 return one/vzero;
218 return -__ieee754_log(fabs(x));
b34b60bc 219 }
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220
221 /* purge negative integers and start evaluation for other x < 0 */
b34b60bc 222 if(hx<0) {
2fedfd5c 223 *signgamp = 1;
b34b60bc 224 if(ix>=0x43300000) /* |x|>=2**52, must be -integer */
2fedfd5c 225 return one/vzero;
b34b60bc 226 t = sin_pi(x);
2fedfd5c 227 if(t==zero) return one/vzero; /* -integer */
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229 if(t<zero) *signgamp = -1;
230 x = -x;
231 }
232
2fedfd5c 233 /* purge 1 and 2 */
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234 if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
235 /* for x < 2.0 */
236 else if(ix<0x40000000) {
237 if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
6ff43c94 238 r = -__ieee754_log(x);
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239 if(ix>=0x3FE76944) {y = one-x; i= 0;}
240 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
241 else {y = x; i=2;}
242 } else {
243 r = zero;
244 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
245 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
246 else {y=x-one;i=2;}
247 }
248 switch(i) {
249 case 0:
250 z = y*y;
251 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
252 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
253 p = y*p1+p2;
2fedfd5c 254 r += p-y/2; break;
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255 case 1:
256 z = y*y;
257 w = z*y;
258 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
259 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
260 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
261 p = z*p1-(tt-w*(p2+y*p3));
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262 r += tf + p; break;
263 case 2:
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264 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
265 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
2fedfd5c 266 r += p1/p2-y/2;
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267 }
268 }
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269 /* x < 8.0 */
270 else if(ix<0x40200000) {
271 i = x;
272 y = x-i;
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273 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
274 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
2fedfd5c 275 r = y/2+p/q;
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276 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
277 switch(i) {
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278 case 7: z *= (y+6); /* FALLTHRU */
279 case 6: z *= (y+5); /* FALLTHRU */
280 case 5: z *= (y+4); /* FALLTHRU */
281 case 4: z *= (y+3); /* FALLTHRU */
282 case 3: z *= (y+2); /* FALLTHRU */
6ff43c94 283 r += __ieee754_log(z); break;
b34b60bc 284 }
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285 /* 8.0 <= x < 2**56 */
286 } else if (ix < 0x43700000) {
6ff43c94 287 t = __ieee754_log(x);
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288 z = one/x;
289 y = z*z;
290 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
291 r = (x-half)*(t-one)+w;
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292 } else
293 /* 2**56 <= x <= inf */
6ff43c94 294 r = x*(__ieee754_log(x)-one);
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295 if(hx<0) r = nadj - r;
296 return r;
297}
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298
299#if (LDBL_MANT_DIG == 53)
300__weak_reference(lgamma_r, lgammal_r);
301#endif