libgo/go/math/lgamma.go

   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 }