1 // Copyright 2010 The Go Authors. All rights reserved. 2 // Use of this source code is governed by a BSD-style 3 // license that can be found in the LICENSE file. 4 5 package math 6 7 /* 8 Floating-point logarithm of the Gamma function. 9 */ 10 11 // The original C code and the long comment below are 12 // from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and 13 // came with this notice. The go code is a simplified 14 // version of the original C. 15 // 16 // ==================================================== 17 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 18 // 19 // Developed at SunPro, a Sun Microsystems, Inc. business. 20 // Permission to use, copy, modify, and distribute this 21 // software is freely granted, provided that this notice 22 // is preserved. 23 // ==================================================== 24 // 25 // __ieee754_lgamma_r(x, signgamp) 26 // Reentrant version of the logarithm of the Gamma function 27 // with user provided pointer for the sign of Gamma(x). 28 // 29 // Method: 30 // 1. Argument Reduction for 0 < x <= 8 31 // Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 32 // reduce x to a number in [1.5,2.5] by 33 // lgamma(1+s) = log(s) + lgamma(s) 34 // for example, 35 // lgamma(7.3) = log(6.3) + lgamma(6.3) 36 // = log(6.3*5.3) + lgamma(5.3) 37 // = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 38 // 2. Polynomial approximation of lgamma around its 39 // minimum (ymin=1.461632144968362245) to maintain monotonicity. 40 // On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 41 // Let z = x-ymin; 42 // lgamma(x) = -1.214862905358496078218 + z**2*poly(z) 43 // poly(z) is a 14 degree polynomial. 44 // 2. Rational approximation in the primary interval [2,3] 45 // We use the following approximation: 46 // s = x-2.0; 47 // lgamma(x) = 0.5*s + s*P(s)/Q(s) 48 // with accuracy 49 // |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 50 // Our algorithms are based on the following observation 51 // 52 // zeta(2)-1 2 zeta(3)-1 3 53 // lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 54 // 2 3 55 // 56 // where Euler = 0.5772156649... is the Euler constant, which 57 // is very close to 0.5. 58 // 59 // 3. For x>=8, we have 60 // lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 61 // (better formula: 62 // lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 63 // Let z = 1/x, then we approximation 64 // f(z) = lgamma(x) - (x-0.5)(log(x)-1) 65 // by 66 // 3 5 11 67 // w = w0 + w1*z + w2*z + w3*z + ... + w6*z 68 // where 69 // |w - f(z)| < 2**-58.74 70 // 71 // 4. For negative x, since (G is gamma function) 72 // -x*G(-x)*G(x) = pi/sin(pi*x), 73 // we have 74 // G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 75 // since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 76 // Hence, for x<0, signgam = sign(sin(pi*x)) and 77 // lgamma(x) = log(|Gamma(x)|) 78 // = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 79 // Note: one should avoid computing pi*(-x) directly in the 80 // computation of sin(pi*(-x)). 81 // 82 // 5. Special Cases 83 // lgamma(2+s) ~ s*(1-Euler) for tiny s 84 // lgamma(1)=lgamma(2)=0 85 // lgamma(x) ~ -log(x) for tiny x 86 // lgamma(0) = lgamma(inf) = inf 87 // lgamma(-integer) = +-inf 88 // 89 // 90 91 var _lgamA = [...]float64{ 92 7.72156649015328655494e-02, // 0x3FB3C467E37DB0C8 93 3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD 94 6.73523010531292681824e-02, // 0x3FB13E001A5562A7 95 2.05808084325167332806e-02, // 0x3F951322AC92547B 96 7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8 97 2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B 98 1.19270763183362067845e-03, // 0x3F538A94116F3F5D 99 5.10069792153511336608e-04, // 0x3F40B6C689B99C00 100 2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D 101 1.08011567247583939954e-04, // 0x3F1C5088987DFB07 102 2.52144565451257326939e-05, // 0x3EFA7074428CFA52 103 4.48640949618915160150e-05, // 0x3F07858E90A45837 104 } 105 var _lgamR = [...]float64{ 106 1.0, // placeholder 107 1.39200533467621045958e+00, // 0x3FF645A762C4AB74 108 7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC 109 1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27 110 1.86459191715652901344e-02, // 0x3F9317EA742ED475 111 7.77942496381893596434e-04, // 0x3F497DDACA41A95B 112 7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140 113 } 114 var _lgamS = [...]float64{ 115 -7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8 116 2.14982415960608852501e-01, // 0x3FCB848B36E20878 117 3.25778796408930981787e-01, // 0x3FD4D98F4F139F59 118 1.46350472652464452805e-01, // 0x3FC2BB9CBEE5F2F7 119 2.66422703033638609560e-02, // 0x3F9B481C7E939961 120 1.84028451407337715652e-03, // 0x3F5E26B67368F239 121 3.19475326584100867617e-05, // 0x3F00BFECDD17E945 122 } 123 var _lgamT = [...]float64{ 124 4.83836122723810047042e-01, // 0x3FDEF72BC8EE38A2 125 -1.47587722994593911752e-01, // 0xBFC2E4278DC6C509 126 6.46249402391333854778e-02, // 0x3FB08B4294D5419B 127 -3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713 128 1.79706750811820387126e-02, // 0x3F9266E7970AF9EC 129 -1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A 130 6.10053870246291332635e-03, // 0x3F78FCE0E370E344 131 -3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7 132 2.25964780900612472250e-03, // 0x3F6282D32E15C915 133 -1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1 134 8.81081882437654011382e-04, // 0x3F4CDF0CEF61A8E9 135 -5.38595305356740546715e-04, // 0xBF41A6109C73E0EC 136 3.15632070903625950361e-04, // 0x3F34AF6D6C0EBBF7 137 -3.12754168375120860518e-04, // 0xBF347F24ECC38C38 138 3.35529192635519073543e-04, // 0x3F35FD3EE8C2D3F4 139 } 140 var _lgamU = [...]float64{ 141 -7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8 142 6.32827064025093366517e-01, // 0x3FE4401E8B005DFF 143 1.45492250137234768737e+00, // 0x3FF7475CD119BD6F 144 9.77717527963372745603e-01, // 0x3FEF497644EA8450 145 2.28963728064692451092e-01, // 0x3FCD4EAEF6010924 146 1.33810918536787660377e-02, // 0x3F8B678BBF2BAB09 147 } 148 var _lgamV = [...]float64{ 149 1.0, 150 2.45597793713041134822e+00, // 0x4003A5D7C2BD619C 151 2.12848976379893395361e+00, // 0x40010725A42B18F5 152 7.69285150456672783825e-01, // 0x3FE89DFBE45050AF 153 1.04222645593369134254e-01, // 0x3FBAAE55D6537C88 154 3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61 155 } 156 var _lgamW = [...]float64{ 157 4.18938533204672725052e-01, // 0x3FDACFE390C97D69 158 8.33333333333329678849e-02, // 0x3FB555555555553B 159 -2.77777777728775536470e-03, // 0xBF66C16C16B02E5C 160 7.93650558643019558500e-04, // 0x3F4A019F98CF38B6 161 -5.95187557450339963135e-04, // 0xBF4380CB8C0FE741 162 8.36339918996282139126e-04, // 0x3F4B67BA4CDAD5D1 163 -1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4 164 } 165 166 // Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x). 167 // 168 // Special cases are: 169 // Lgamma(+Inf) = +Inf 170 // Lgamma(0) = +Inf 171 // Lgamma(-integer) = +Inf 172 // Lgamma(-Inf) = -Inf 173 // Lgamma(NaN) = NaN 174 func Lgamma(x float64) (lgamma float64, sign int) { 175 const ( 176 Ymin = 1.461632144968362245 177 Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15 178 Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15 179 Two58 = 1 << 58 // 0x4390000000000000 ~2.8823e+17 180 Tiny = 1.0 / (1 << 70) // 0x3b90000000000000 ~8.47033e-22 181 Tc = 1.46163214496836224576e+00 // 0x3FF762D86356BE3F 182 Tf = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42 183 // Tt = -(tail of Tf) 184 Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F 185 ) 186 // special cases 187 sign = 1 188 switch { 189 case IsNaN(x): 190 lgamma = x 191 return 192 case IsInf(x, 0): 193 lgamma = x 194 return 195 case x == 0: 196 lgamma = Inf(1) 197 return 198 } 199 200 neg := false 201 if x < 0 { 202 x = -x 203 neg = true 204 } 205 206 if x < Tiny { // if |x| < 2**-70, return -log(|x|) 207 if neg { 208 sign = -1 209 } 210 lgamma = -Log(x) 211 return 212 } 213 var nadj float64 214 if neg { 215 if x >= Two52 { // |x| >= 2**52, must be -integer 216 lgamma = Inf(1) 217 return 218 } 219 t := sinPi(x) 220 if t == 0 { 221 lgamma = Inf(1) // -integer 222 return 223 } 224 nadj = Log(Pi / Abs(t*x)) 225 if t < 0 { 226 sign = -1 227 } 228 } 229 230 switch { 231 case x == 1 || x == 2: // purge off 1 and 2 232 lgamma = 0 233 return 234 case x < 2: // use lgamma(x) = lgamma(x+1) - log(x) 235 var y float64 236 var i int 237 if x <= 0.9 { 238 lgamma = -Log(x) 239 switch { 240 case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <= 0.9 241 y = 1 - x 242 i = 0 243 case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316 244 y = x - (Tc - 1) 245 i = 1 246 default: // 0 < x < 0.2316 247 y = x 248 i = 2 249 } 250 } else { 251 lgamma = 0 252 switch { 253 case x >= (Ymin + 0.27): // 1.7316 <= x < 2 254 y = 2 - x 255 i = 0 256 case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316 257 y = x - Tc 258 i = 1 259 default: // 0.9 < x < 1.2316 260 y = x - 1 261 i = 2 262 } 263 } 264 switch i { 265 case 0: 266 z := y * y 267 p1 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10])))) 268 p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11]))))) 269 p := y*p1 + p2 270 lgamma += (p - 0.5*y) 271 case 1: 272 z := y * y 273 w := z * y 274 p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp 275 p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13]))) 276 p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14]))) 277 p := z*p1 - (Tt - w*(p2+y*p3)) 278 lgamma += (Tf + p) 279 case 2: 280 p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5]))))) 281 p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5])))) 282 lgamma += (-0.5*y + p1/p2) 283 } 284 case x < 8: // 2 <= x < 8 285 i := int(x) 286 y := x - float64(i) 287 p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6])))))) 288 q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6]))))) 289 lgamma = 0.5*y + p/q 290 z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s) 291 switch i { 292 case 7: 293 z *= (y + 6) 294 fallthrough 295 case 6: 296 z *= (y + 5) 297 fallthrough 298 case 5: 299 z *= (y + 4) 300 fallthrough 301 case 4: 302 z *= (y + 3) 303 fallthrough 304 case 3: 305 z *= (y + 2) 306 lgamma += Log(z) 307 } 308 case x < Two58: // 8 <= x < 2**58 309 t := Log(x) 310 z := 1 / x 311 y := z * z 312 w := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6]))))) 313 lgamma = (x-0.5)*(t-1) + w 314 default: // 2**58 <= x <= Inf 315 lgamma = x * (Log(x) - 1) 316 } 317 if neg { 318 lgamma = nadj - lgamma 319 } 320 return 321 } 322 323 // sinPi(x) is a helper function for negative x 324 func sinPi(x float64) float64 { 325 const ( 326 Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15 327 Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15 328 ) 329 if x < 0.25 { 330 return -Sin(Pi * x) 331 } 332 333 // argument reduction 334 z := Floor(x) 335 var n int 336 if z != x { // inexact 337 x = Mod(x, 2) 338 n = int(x * 4) 339 } else { 340 if x >= Two53 { // x must be even 341 x = 0 342 n = 0 343 } else { 344 if x < Two52 { 345 z = x + Two52 // exact 346 } 347 n = int(1 & Float64bits(z)) 348 x = float64(n) 349 n <<= 2 350 } 351 } 352 switch n { 353 case 0: 354 x = Sin(Pi * x) 355 case 1, 2: 356 x = Cos(Pi * (0.5 - x)) 357 case 3, 4: 358 x = Sin(Pi * (1 - x)) 359 case 5, 6: 360 x = -Cos(Pi * (x - 1.5)) 361 default: 362 x = Sin(Pi * (x - 2)) 363 } 364 return -x 365 } 366
