math: add lgamma; in-line special cases of acosh, nextafter

Added lgamma.go, tests and special cases.

R=rsc
CC=golang-dev
https://golang.org/cl/217060

diff --git a/src/pkg/math/Makefile b/src/pkg/math/Makefile
index e8c42529388767b107779cdc78d73cbfa16115ca..e24c448f88d3d7f06606e118e3b52a643911a4ea 100644 (file)
--- a/src/pkg/math/Makefile
+++ b/src/pkg/math/Makefile
@@ -52,6 +52,7 @@ ALLGOFILES=\
        fmod.go\
        frexp.go\
        hypot.go\
+       lgamma.go\
        ldexp.go\
        log.go\
        log1p.go\

diff --git a/src/pkg/math/acosh.go b/src/pkg/math/acosh.go
index 1f0d3f338047d601e6018194d45011429f751387..13afc5aec2cc2d312d4c71649b2c6e6cf21d715e 100644 (file)
--- a/src/pkg/math/acosh.go
+++ b/src/pkg/math/acosh.go
@@ -44,8 +44,11 @@ func Acosh(x float64) float64 {
                Ln2   = 6.93147180559945286227e-01 // 0x3FE62E42FEFA39EF
                Large = 1 << 28                    // 2^28
        )
+       // TODO(rsc): Remove manual inlining of IsNaN
+       // when compiler does it for us
+       // first case is special case
        switch {
-       case x < 1 || IsNaN(x):
+       case x < 1 || x != x: // x < 1 || IsNaN(x):
                return NaN()
        case x == 1:
                return 0

diff --git a/src/pkg/math/all_test.go b/src/pkg/math/all_test.go
index 1109165280d5af6dc8fe0d819e1d588b925920dc..ef4806540a9de658bcae44df87ad492c142af45b 100644 (file)
--- a/src/pkg/math/all_test.go
+++ b/src/pkg/math/all_test.go
@@ -286,7 +286,18 @@ var frexp = []fi{
        fi{9.1265404584042750000e-01, 1},
        fi{-5.4287029803597508250e-01, 4},
 }
-
+var lgamma = []fi{
+       fi{3.146492141244545774319734e+00, 1},
+       fi{8.003414490659126375852113e+00, 1},
+       fi{1.517575735509779707488106e+00, -1},
+       fi{-2.588480028182145853558748e-01, 1},
+       fi{1.1989897050205555002007985e+01, 1},
+       fi{6.262899811091257519386906e-01, 1},
+       fi{3.5287924899091566764846037e+00, 1},
+       fi{4.5725644770161182299423372e-01, 1},
+       fi{-6.363667087767961257654854e-02, 1},
+       fi{-1.077385130910300066425564e+01, -1},
+}
 var log = []float64{
        1.605231462693062999102599e+00,
        2.0462560018708770653153909e+00,
@@ -736,6 +747,21 @@ var hypotSC = []float64{
        NaN(),
 }
 
+var vflgammaSC = []float64{
+       Inf(-1),
+       -3,
+       0,
+       Inf(1),
+       NaN(),
+}
+var lgammaSC = []fi{
+       fi{Inf(-1), 1},
+       fi{Inf(1), 1},
+       fi{Inf(1), 1},
+       fi{Inf(1), 1},
+       fi{NaN(), 1},
+}
+
 var vflogSC = []float64{
        Inf(-1),
        -Pi,
@@ -1229,6 +1255,19 @@ func TestLdexp(t *testing.T) {
        }
 }
 
+func TestLgamma(t *testing.T) {
+       for i := 0; i < len(vf); i++ {
+               if f, s := Lgamma(vf[i]); !close(lgamma[i].f, f) || lgamma[i].i != s {
+                       t.Errorf("Lgamma(%g) = %g, %d, want %g, %d\n", vf[i], f, s, lgamma[i].f, lgamma[i].i)
+               }
+       }
+       for i := 0; i < len(vflgammaSC); i++ {
+               if f, s := Lgamma(vflgammaSC[i]); !alike(lgammaSC[i].f, f) || lgammaSC[i].i != s {
+                       t.Errorf("Lgamma(%g) = %g, %d, want %g, %d\n", vflgammaSC[i], f, s, lgammaSC[i].f, lgammaSC[i].i)
+               }
+       }
+}
+
 func TestLog(t *testing.T) {
        for i := 0; i < len(vf); i++ {
                a := Fabs(vf[i])
@@ -1632,6 +1671,12 @@ func BenchmarkLdexp(b *testing.B) {
        }
 }
 
+func BenchmarkLgamma(b *testing.B) {
+       for i := 0; i < b.N; i++ {
+               Lgamma(2.5)
+       }
+}
+
 func BenchmarkLog(b *testing.B) {
        for i := 0; i < b.N; i++ {
                Log(.5)

diff --git a/src/pkg/math/lgamma.go b/src/pkg/math/lgamma.go
new file mode 100644 (file)
index 0000000..38558f3
--- /dev/null
+++ b/src/pkg/math/lgamma.go
@@ -0,0 +1,350 @@
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+       Floating-point logarithm of the Gamma function.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and
+// came with this notice.  The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_lgamma_r(x, signgamp)
+// Reentrant version of the logarithm of the Gamma function
+// with user provided pointer for the sign of Gamma(x).
+//
+// Method:
+//   1. Argument Reduction for 0 < x <= 8
+//      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
+//      reduce x to a number in [1.5,2.5] by
+//              lgamma(1+s) = log(s) + lgamma(s)
+//      for example,
+//              lgamma(7.3) = log(6.3) + lgamma(6.3)
+//                          = log(6.3*5.3) + lgamma(5.3)
+//                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
+//   2. Polynomial approximation of lgamma around its
+//      minimum (ymin=1.461632144968362245) to maintain monotonicity.
+//      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
+//              Let z = x-ymin;
+//              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
+//              poly(z) is a 14 degree polynomial.
+//   2. Rational approximation in the primary interval [2,3]
+//      We use the following approximation:
+//              s = x-2.0;
+//              lgamma(x) = 0.5*s + s*P(s)/Q(s)
+//      with accuracy
+//              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
+//      Our algorithms are based on the following observation
+//
+//                             zeta(2)-1    2    zeta(3)-1    3
+// lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
+//                                 2                 3
+//
+//      where Euler = 0.5772156649... is the Euler constant, which
+//      is very close to 0.5.
+//
+//   3. For x>=8, we have
+//      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
+//      (better formula:
+//         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
+//      Let z = 1/x, then we approximation
+//              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
+//      by
+//                                  3       5             11
+//              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
+//      where
+//              |w - f(z)| < 2**-58.74
+//
+//   4. For negative x, since (G is gamma function)
+//              -x*G(-x)*G(x) = pi/sin(pi*x),
+//      we have
+//              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
+//      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
+//      Hence, for x<0, signgam = sign(sin(pi*x)) and
+//              lgamma(x) = log(|Gamma(x)|)
+//                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
+//      Note: one should avoid computing pi*(-x) directly in the
+//            computation of sin(pi*(-x)).
+//
+//   5. Special Cases
+//              lgamma(2+s) ~ s*(1-Euler) for tiny s
+//              lgamma(1)=lgamma(2)=0
+//              lgamma(x) ~ -log(x) for tiny x
+//              lgamma(0) = lgamma(inf) = inf
+//              lgamma(-integer) = +-inf
+//
+//
+
+// Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
+//
+// Special cases are:
+//     Lgamma(+Inf) = +Inf
+//     Lgamma(0) = +Inf
+//     Lgamma(-integer) = +Inf
+//     Lgamma(-Inf) = -Inf
+//     Lgamma(NaN) = NaN
+func Lgamma(x float64) (lgamma float64, sign int) {
+       const (
+               Ymin  = 1.461632144968362245
+               Two52 = 1 << 52                     // 0x4330000000000000 ~4.5036e+15
+               Two53 = 1 << 53                     // 0x4340000000000000 ~9.0072e+15
+               Two58 = 1 << 58                     // 0x4390000000000000 ~2.8823e+17
+               Tiny  = 1.0 / (1 << 70)             // 0x3b90000000000000 ~8.47033e-22
+               A0    = 7.72156649015328655494e-02  // 0x3FB3C467E37DB0C8
+               A1    = 3.22467033424113591611e-01  // 0x3FD4A34CC4A60FAD
+               A2    = 6.73523010531292681824e-02  // 0x3FB13E001A5562A7
+               A3    = 2.05808084325167332806e-02  // 0x3F951322AC92547B
+               A4    = 7.38555086081402883957e-03  // 0x3F7E404FB68FEFE8
+               A5    = 2.89051383673415629091e-03  // 0x3F67ADD8CCB7926B
+               A6    = 1.19270763183362067845e-03  // 0x3F538A94116F3F5D
+               A7    = 5.10069792153511336608e-04  // 0x3F40B6C689B99C00
+               A8    = 2.20862790713908385557e-04  // 0x3F2CF2ECED10E54D
+               A9    = 1.08011567247583939954e-04  // 0x3F1C5088987DFB07
+               A10   = 2.52144565451257326939e-05  // 0x3EFA7074428CFA52
+               A11   = 4.48640949618915160150e-05  // 0x3F07858E90A45837
+               Tc    = 1.46163214496836224576e+00  // 0x3FF762D86356BE3F
+               Tf    = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
+               // Tt = -(tail of Tf)
+               Tt  = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
+               T0  = 4.83836122723810047042e-01  // 0x3FDEF72BC8EE38A2
+               T1  = -1.47587722994593911752e-01 // 0xBFC2E4278DC6C509
+               T2  = 6.46249402391333854778e-02  // 0x3FB08B4294D5419B
+               T3  = -3.27885410759859649565e-02 // 0xBFA0C9A8DF35B713
+               T4  = 1.79706750811820387126e-02  // 0x3F9266E7970AF9EC
+               T5  = -1.03142241298341437450e-02 // 0xBF851F9FBA91EC6A
+               T6  = 6.10053870246291332635e-03  // 0x3F78FCE0E370E344
+               T7  = -3.68452016781138256760e-03 // 0xBF6E2EFFB3E914D7
+               T8  = 2.25964780900612472250e-03  // 0x3F6282D32E15C915
+               T9  = -1.40346469989232843813e-03 // 0xBF56FE8EBF2D1AF1
+               T10 = 8.81081882437654011382e-04  // 0x3F4CDF0CEF61A8E9
+               T11 = -5.38595305356740546715e-04 // 0xBF41A6109C73E0EC
+               T12 = 3.15632070903625950361e-04  // 0x3F34AF6D6C0EBBF7
+               T13 = -3.12754168375120860518e-04 // 0xBF347F24ECC38C38
+               T14 = 3.35529192635519073543e-04  // 0x3F35FD3EE8C2D3F4
+               U0  = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
+               U1  = 6.32827064025093366517e-01  // 0x3FE4401E8B005DFF
+               U2  = 1.45492250137234768737e+00  // 0x3FF7475CD119BD6F
+               U3  = 9.77717527963372745603e-01  // 0x3FEF497644EA8450
+               U4  = 2.28963728064692451092e-01  // 0x3FCD4EAEF6010924
+               U5  = 1.33810918536787660377e-02  // 0x3F8B678BBF2BAB09
+               V1  = 2.45597793713041134822e+00  // 0x4003A5D7C2BD619C
+               V2  = 2.12848976379893395361e+00  // 0x40010725A42B18F5
+               V3  = 7.69285150456672783825e-01  // 0x3FE89DFBE45050AF
+               V4  = 1.04222645593369134254e-01  // 0x3FBAAE55D6537C88
+               V5  = 3.21709242282423911810e-03  // 0x3F6A5ABB57D0CF61
+               S0  = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
+               S1  = 2.14982415960608852501e-01  // 0x3FCB848B36E20878
+               S2  = 3.25778796408930981787e-01  // 0x3FD4D98F4F139F59
+               S3  = 1.46350472652464452805e-01  // 0x3FC2BB9CBEE5F2F7
+               S4  = 2.66422703033638609560e-02  // 0x3F9B481C7E939961
+               S5  = 1.84028451407337715652e-03  // 0x3F5E26B67368F239
+               S6  = 3.19475326584100867617e-05  // 0x3F00BFECDD17E945
+               R1  = 1.39200533467621045958e+00  // 0x3FF645A762C4AB74
+               R2  = 7.21935547567138069525e-01  // 0x3FE71A1893D3DCDC
+               R3  = 1.71933865632803078993e-01  // 0x3FC601EDCCFBDF27
+               R4  = 1.86459191715652901344e-02  // 0x3F9317EA742ED475
+               R5  = 7.77942496381893596434e-04  // 0x3F497DDACA41A95B
+               R6  = 7.32668430744625636189e-06  // 0x3EDEBAF7A5B38140
+               W0  = 4.18938533204672725052e-01  // 0x3FDACFE390C97D69
+               W1  = 8.33333333333329678849e-02  // 0x3FB555555555553B
+               W2  = -2.77777777728775536470e-03 // 0xBF66C16C16B02E5C
+               W3  = 7.93650558643019558500e-04  // 0x3F4A019F98CF38B6
+               W4  = -5.95187557450339963135e-04 // 0xBF4380CB8C0FE741
+               W5  = 8.36339918996282139126e-04  // 0x3F4B67BA4CDAD5D1
+               W6  = -1.63092934096575273989e-03 // 0xBF5AB89D0B9E43E4
+       )
+       // TODO(rsc): Remove manual inlining of IsNaN, IsInf
+       // when compiler does it for us
+       // special cases
+       sign = 1
+       switch {
+       case x != x: // IsNaN(x):
+               lgamma = x
+               return
+       case x < -MaxFloat64 || x > MaxFloat64: // IsInf(x, 0):
+               lgamma = x
+               return
+       case x == 0:
+               lgamma = Inf(1)
+               return
+       }
+
+       neg := false
+       if x < 0 {
+               x = -x
+               neg = true
+       }
+
+       if x < Tiny { // if |x| < 2**-70, return -log(|x|)
+               if neg {
+                       sign = -1
+               }
+               lgamma = -Log(x)
+               return
+       }
+       var nadj float64
+       if neg {
+               if x >= Two52 { // |x| >= 2**52, must be -integer
+                       lgamma = Inf(1)
+                       return
+               }
+               t := sinPi(x)
+               if t == 0 {
+                       lgamma = Inf(1) // -integer
+                       return
+               }
+               nadj = Log(Pi / Fabs(t*x))
+               if t < 0 {
+                       sign = -1
+               }
+       }
+
+       switch {
+       case x == 1 || x == 2: // purge off 1 and 2
+               lgamma = 0
+               return
+       case x < 2: // use lgamma(x) = lgamma(x+1) - log(x)
+               var y float64
+               var i int
+               if x <= 0.9 {
+                       lgamma = -Log(x)
+                       switch {
+                       case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <=  0.9
+                               y = 1 - x
+                               i = 0
+                       case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316
+                               y = x - (Tc - 1)
+                               i = 1
+                       default: // 0 < x < 0.2316
+                               y = x
+                               i = 2
+                       }
+               } else {
+                       lgamma = 0
+                       switch {
+                       case x >= (Ymin + 0.27): // 1.7316 <= x < 2
+                               y = 2 - x
+                               i = 0
+                       case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316
+                               y = x - Tc
+                               i = 1
+                       default: // 0.9 < x < 1.2316
+                               y = x - 1
+                               i = 2
+                       }
+               }
+               switch i {
+               case 0:
+                       z := y * y
+                       p1 := A0 + z*(A2+z*(A4+z*(A6+z*(A8+z*A10))))
+                       p2 := z * (A1 + z*(A3+z*(A5+z*(A7+z*(A9+z*A11)))))
+                       p := y*p1 + p2
+                       lgamma += (p - 0.5*y)
+               case 1:
+                       z := y * y
+                       w := z * y
+                       p1 := T0 + w*(T3+w*(T6+w*(T9+w*T12))) // parallel comp
+                       p2 := T1 + w*(T4+w*(T7+w*(T10+w*T13)))
+                       p3 := T2 + w*(T5+w*(T8+w*(T11+w*T14)))
+                       p := z*p1 - (Tt - w*(p2+y*p3))
+                       lgamma += (Tf + p)
+               case 2:
+                       p1 := y * (U0 + y*(U1+y*(U2+y*(U3+y*(U4+y*U5)))))
+                       p2 := 1 + y*(V1+y*(V2+y*(V3+y*(V4+y*V5))))
+                       lgamma += (-0.5*y + p1/p2)
+               }
+       case x < 8: // 2 <= x < 8
+               i := int(x)
+               y := x - float64(i)
+               p := y * (S0 + y*(S1+y*(S2+y*(S3+y*(S4+y*(S5+y*S6))))))
+               q := 1 + y*(R1+y*(R2+y*(R3+y*(R4+y*(R5+y*R6)))))
+               lgamma = 0.5*y + p/q
+               z := float64(1) // Lgamma(1+s) = Log(s) + Lgamma(s)
+               switch i {
+               case 7:
+                       z *= (y + 6)
+                       fallthrough
+               case 6:
+                       z *= (y + 5)
+                       fallthrough
+               case 5:
+                       z *= (y + 4)
+                       fallthrough
+               case 4:
+                       z *= (y + 3)
+                       fallthrough
+               case 3:
+                       z *= (y + 2)
+                       lgamma += Log(z)
+               }
+       case x < Two58: // 8 <= x < 2**58
+               t := Log(x)
+               z := 1 / x
+               y := z * z
+               w := W0 + z*(W1+y*(W2+y*(W3+y*(W4+y*(W5+y*W6)))))
+               lgamma = (x-0.5)*(t-1) + w
+       default: // 2**58 <= x <= Inf
+               lgamma = x * (Log(x) - 1)
+       }
+       if neg {
+               lgamma = nadj - lgamma
+       }
+       return
+}
+
+// sinPi(x) is a helper function for negative x
+func sinPi(x float64) float64 {
+       const (
+               Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
+               Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
+       )
+       if x < 0.25 {
+               return -Sin(Pi * x)
+       }
+
+       // argument reduction
+       z := Floor(x)
+       var n int
+       if z != x { // inexact
+               x = Fmod(x, 2)
+               n = int(x * 4)
+       } else {
+               if x >= Two53 { // x must be even
+                       x = 0
+                       n = 0
+               } else {
+                       if x < Two52 {
+                               z = x + Two52 // exact
+                       }
+                       n = int(1 & Float64bits(z))
+                       x = float64(n)
+                       n <<= 2
+               }
+       }
+       switch n {
+       case 0:
+               x = Sin(Pi * x)
+       case 1, 2:
+               x = Cos(Pi * (0.5 - x))
+       case 3, 4:
+               x = Sin(Pi * (1 - x))
+       case 5, 6:
+               x = -Cos(Pi * (x - 1.5))
+       default:
+               x = Sin(Pi * (x - 2))
+       }
+       return -x
+}

diff --git a/src/pkg/math/nextafter.go b/src/pkg/math/nextafter.go
index b57d3e715983bb79a55d5adf8a3ab1b28c69846c..86114340c1eaacc592d794a2ff813ff51f984891 100644 (file)
--- a/src/pkg/math/nextafter.go
+++ b/src/pkg/math/nextafter.go
@@ -11,8 +11,10 @@ package math
 //     Nextafter(NaN, y) = NaN
 //     Nextafter(x, NaN) = NaN
 func Nextafter(x, y float64) (r float64) {
+       // TODO(rsc): Remove manual inlining of IsNaN
+       // when compiler does it for us
        switch {
-       case IsNaN(x) || IsNaN(y): // special case
+       case x != x || y != y: // IsNaN(x) || IsNaN(y): // special case
                r = NaN()
        case x == y:
                r = x