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 // Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
  92 //
  93 // Special cases are:
  94 //      Lgamma(+Inf) = +Inf
  95 //      Lgamma(0) = +Inf
  96 //      Lgamma(-integer) = +Inf
  97 //      Lgamma(-Inf) = -Inf
  98 //      Lgamma(NaN) = NaN
  99 func Lgamma(x float64) (lgamma float64, sign int) {
 100         const (
 101                 Ymin  = 1.461632144968362245
 102                 Two52 = 1 << 52                     // 0x4330000000000000 ~4.5036e+15
 103                 Two53 = 1 << 53                     // 0x4340000000000000 ~9.0072e+15
 104                 Two58 = 1 << 58                     // 0x4390000000000000 ~2.8823e+17
 105                 Tiny  = 1.0 / (1 << 70)             // 0x3b90000000000000 ~8.47033e-22
 106                 A0    = 7.72156649015328655494e-02  // 0x3FB3C467E37DB0C8
 107                 A1    = 3.22467033424113591611e-01  // 0x3FD4A34CC4A60FAD
 108                 A2    = 6.73523010531292681824e-02  // 0x3FB13E001A5562A7
 109                 A3    = 2.05808084325167332806e-02  // 0x3F951322AC92547B
 110                 A4    = 7.38555086081402883957e-03  // 0x3F7E404FB68FEFE8
 111                 A5    = 2.89051383673415629091e-03  // 0x3F67ADD8CCB7926B
 112                 A6    = 1.19270763183362067845e-03  // 0x3F538A94116F3F5D
 113                 A7    = 5.10069792153511336608e-04  // 0x3F40B6C689B99C00
 114                 A8    = 2.20862790713908385557e-04  // 0x3F2CF2ECED10E54D
 115                 A9    = 1.08011567247583939954e-04  // 0x3F1C5088987DFB07
 116                 A10   = 2.52144565451257326939e-05  // 0x3EFA7074428CFA52
 117                 A11   = 4.48640949618915160150e-05  // 0x3F07858E90A45837
 118                 Tc    = 1.46163214496836224576e+00  // 0x3FF762D86356BE3F
 119                 Tf    = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
 120                 // Tt = -(tail of Tf)
 121                 Tt  = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
 122                 T0  = 4.83836122723810047042e-01  // 0x3FDEF72BC8EE38A2
 123                 T1  = -1.47587722994593911752e-01 // 0xBFC2E4278DC6C509
 124                 T2  = 6.46249402391333854778e-02  // 0x3FB08B4294D5419B
 125                 T3  = -3.27885410759859649565e-02 // 0xBFA0C9A8DF35B713
 126                 T4  = 1.79706750811820387126e-02  // 0x3F9266E7970AF9EC
 127                 T5  = -1.03142241298341437450e-02 // 0xBF851F9FBA91EC6A
 128                 T6  = 6.10053870246291332635e-03  // 0x3F78FCE0E370E344
 129                 T7  = -3.68452016781138256760e-03 // 0xBF6E2EFFB3E914D7
 130                 T8  = 2.25964780900612472250e-03  // 0x3F6282D32E15C915
 131                 T9  = -1.40346469989232843813e-03 // 0xBF56FE8EBF2D1AF1
 132                 T10 = 8.81081882437654011382e-04  // 0x3F4CDF0CEF61A8E9
 133                 T11 = -5.38595305356740546715e-04 // 0xBF41A6109C73E0EC
 134                 T12 = 3.15632070903625950361e-04  // 0x3F34AF6D6C0EBBF7
 135                 T13 = -3.12754168375120860518e-04 // 0xBF347F24ECC38C38
 136                 T14 = 3.35529192635519073543e-04  // 0x3F35FD3EE8C2D3F4
 137                 U0  = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
 138                 U1  = 6.32827064025093366517e-01  // 0x3FE4401E8B005DFF
 139                 U2  = 1.45492250137234768737e+00  // 0x3FF7475CD119BD6F
 140                 U3  = 9.77717527963372745603e-01  // 0x3FEF497644EA8450
 141                 U4  = 2.28963728064692451092e-01  // 0x3FCD4EAEF6010924
 142                 U5  = 1.33810918536787660377e-02  // 0x3F8B678BBF2BAB09
 143                 V1  = 2.45597793713041134822e+00  // 0x4003A5D7C2BD619C
 144                 V2  = 2.12848976379893395361e+00  // 0x40010725A42B18F5
 145                 V3  = 7.69285150456672783825e-01  // 0x3FE89DFBE45050AF
 146                 V4  = 1.04222645593369134254e-01  // 0x3FBAAE55D6537C88
 147                 V5  = 3.21709242282423911810e-03  // 0x3F6A5ABB57D0CF61
 148                 S0  = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
 149                 S1  = 2.14982415960608852501e-01  // 0x3FCB848B36E20878
 150                 S2  = 3.25778796408930981787e-01  // 0x3FD4D98F4F139F59
 151                 S3  = 1.46350472652464452805e-01  // 0x3FC2BB9CBEE5F2F7
 152                 S4  = 2.66422703033638609560e-02  // 0x3F9B481C7E939961
 153                 S5  = 1.84028451407337715652e-03  // 0x3F5E26B67368F239
 154                 S6  = 3.19475326584100867617e-05  // 0x3F00BFECDD17E945
 155                 R1  = 1.39200533467621045958e+00  // 0x3FF645A762C4AB74
 156                 R2  = 7.21935547567138069525e-01  // 0x3FE71A1893D3DCDC
 157                 R3  = 1.71933865632803078993e-01  // 0x3FC601EDCCFBDF27
 158                 R4  = 1.86459191715652901344e-02  // 0x3F9317EA742ED475
 159                 R5  = 7.77942496381893596434e-04  // 0x3F497DDACA41A95B
 160                 R6  = 7.32668430744625636189e-06  // 0x3EDEBAF7A5B38140
 161                 W0  = 4.18938533204672725052e-01  // 0x3FDACFE390C97D69
 162                 W1  = 8.33333333333329678849e-02  // 0x3FB555555555553B
 163                 W2  = -2.77777777728775536470e-03 // 0xBF66C16C16B02E5C
 164                 W3  = 7.93650558643019558500e-04  // 0x3F4A019F98CF38B6
 165                 W4  = -5.95187557450339963135e-04 // 0xBF4380CB8C0FE741
 166                 W5  = 8.36339918996282139126e-04  // 0x3F4B67BA4CDAD5D1
 167                 W6  = -1.63092934096575273989e-03 // 0xBF5AB89D0B9E43E4
 168         )
 169         // TODO(rsc): Remove manual inlining of IsNaN, IsInf
 170         // when compiler does it for us
 171         // special cases
 172         sign = 1
 173         switch {
 174         case x != x: // IsNaN(x):
 175                 lgamma = x
 176                 return
 177         case x < -MaxFloat64 || x > MaxFloat64: // IsInf(x, 0):
 178                 lgamma = x
 179                 return
 180         case x == 0:
 181                 lgamma = Inf(1)
 182                 return
 183         }
 184
 185         neg := false
 186         if x < 0 {
 187                 x = -x
 188                 neg = true
 189         }
 190
 191         if x < Tiny { // if |x| < 2**-70, return -log(|x|)
 192                 if neg {
 193                         sign = -1
 194                 }
 195                 lgamma = -Log(x)
 196                 return
 197         }
 198         var nadj float64
 199         if neg {
 200                 if x >= Two52 { // |x| >= 2**52, must be -integer
 201                         lgamma = Inf(1)
 202                         return
 203                 }
 204                 t := sinPi(x)
 205                 if t == 0 {
 206                         lgamma = Inf(1) // -integer
 207                         return
 208                 }
 209                 nadj = Log(Pi / Abs(t*x))
 210                 if t < 0 {
 211                         sign = -1
 212                 }
 213         }
 214
 215         switch {
 216         case x == 1 || x == 2: // purge off 1 and 2
 217                 lgamma = 0
 218                 return
 219         case x < 2: // use lgamma(x) = lgamma(x+1) - log(x)
 220                 var y float64
 221                 var i int
 222                 if x <= 0.9 {
 223                         lgamma = -Log(x)
 224                         switch {
 225                         case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <=  0.9
 226                                 y = 1 - x
 227                                 i = 0
 228                         case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316
 229                                 y = x - (Tc - 1)
 230                                 i = 1
 231                         default: // 0 < x < 0.2316
 232                                 y = x
 233                                 i = 2
 234                         }
 235                 } else {
 236                         lgamma = 0
 237                         switch {
 238                         case x >= (Ymin + 0.27): // 1.7316 <= x < 2
 239                                 y = 2 - x
 240                                 i = 0
 241                         case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316
 242                                 y = x - Tc
 243                                 i = 1
 244                         default: // 0.9 < x < 1.2316
 245                                 y = x - 1
 246                                 i = 2
 247                         }
 248                 }
 249                 switch i {
 250                 case 0:
 251                         z := y * y
 252                         p1 := A0 + z*(A2+z*(A4+z*(A6+z*(A8+z*A10))))
 253                         p2 := z * (A1 + z*(A3+z*(A5+z*(A7+z*(A9+z*A11)))))
 254                         p := y*p1 + p2
 255                         lgamma += (p - 0.5*y)
 256                 case 1:
 257                         z := y * y
 258                         w := z * y
 259                         p1 := T0 + w*(T3+w*(T6+w*(T9+w*T12))) // parallel comp
 260                         p2 := T1 + w*(T4+w*(T7+w*(T10+w*T13)))
 261                         p3 := T2 + w*(T5+w*(T8+w*(T11+w*T14)))
 262                         p := z*p1 - (Tt - w*(p2+y*p3))
 263                         lgamma += (Tf + p)
 264                 case 2:
 265                         p1 := y * (U0 + y*(U1+y*(U2+y*(U3+y*(U4+y*U5)))))
 266                         p2 := 1 + y*(V1+y*(V2+y*(V3+y*(V4+y*V5))))
 267                         lgamma += (-0.5*y + p1/p2)
 268                 }
 269         case x < 8: // 2 <= x < 8
 270                 i := int(x)
 271                 y := x - float64(i)
 272                 p := y * (S0 + y*(S1+y*(S2+y*(S3+y*(S4+y*(S5+y*S6))))))
 273                 q := 1 + y*(R1+y*(R2+y*(R3+y*(R4+y*(R5+y*R6)))))
 274                 lgamma = 0.5*y + p/q
 275                 z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)
 276                 switch i {
 277                 case 7:
 278                         z *= (y + 6)
 279                         fallthrough
 280                 case 6:
 281                         z *= (y + 5)
 282                         fallthrough
 283                 case 5:
 284                         z *= (y + 4)
 285                         fallthrough
 286                 case 4:
 287                         z *= (y + 3)
 288                         fallthrough
 289                 case 3:
 290                         z *= (y + 2)
 291                         lgamma += Log(z)
 292                 }
 293         case x < Two58: // 8 <= x < 2**58
 294                 t := Log(x)
 295                 z := 1 / x
 296                 y := z * z
 297                 w := W0 + z*(W1+y*(W2+y*(W3+y*(W4+y*(W5+y*W6)))))
 298                 lgamma = (x-0.5)*(t-1) + w
 299         default: // 2**58 <= x <= Inf
 300                 lgamma = x * (Log(x) - 1)
 301         }
 302         if neg {
 303                 lgamma = nadj - lgamma
 304         }
 305         return
 306 }
 307
 308 // sinPi(x) is a helper function for negative x
 309 func sinPi(x float64) float64 {
 310         const (
 311                 Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
 312                 Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
 313         )
 314         if x < 0.25 {
 315                 return -Sin(Pi * x)
 316         }
 317
 318         // argument reduction
 319         z := Floor(x)
 320         var n int
 321         if z != x { // inexact
 322                 x = Mod(x, 2)
 323                 n = int(x * 4)
 324         } else {
 325                 if x >= Two53 { // x must be even
 326                         x = 0
 327                         n = 0
 328                 } else {
 329                         if x < Two52 {
 330                                 z = x + Two52 // exact
 331                         }
 332                         n = int(1 & Float64bits(z))
 333                         x = float64(n)
 334                         n <<= 2
 335                 }
 336         }
 337         switch n {
 338         case 0:
 339                 x = Sin(Pi * x)
 340         case 1, 2:
 341                 x = Cos(Pi * (0.5 - x))
 342         case 3, 4:
 343                 x = Sin(Pi * (1 - x))
 344         case 5, 6:
 345                 x = -Cos(Pi * (x - 1.5))
 346         default:
 347                 x = Sin(Pi * (x - 2))
 348         }
 349         return -x
 350 }