math: add Gamma function

R=rsc
CC=golang-dev
https://golang.org/cl/649041

src/pkg/math/Makefile

@@ -53,6 +53,7 @@ ALLGOFILES=\
 	floor.go\
 	fmod.go\
 	frexp.go\
+	gamma.go\
 	hypot.go\
 	hypot_port.go\
 	logb.go\

src/pkg/math/all_test.go

@@ -286,6 +286,18 @@ var frexp = []fi{
 	fi{9.1265404584042750000e-01, 1},
 	fi{-5.4287029803597508250e-01, 4},
 }
+var gamma = []float64{
+	2.3254348370739963835386613898e+01,
+	2.991153837155317076427529816e+03,
+	-4.561154336726758060575129109e+00,
+	7.719403468842639065959210984e-01,
+	1.6111876618855418534325755566e+05,
+	1.8706575145216421164173224946e+00,
+	3.4082787447257502836734201635e+01,
+	1.579733951448952054898583387e+00,
+	9.3834586598354592860187267089e-01,
+	-2.093995902923148389186189429e-05,
+}
 var lgamma = []fi{
 	fi{3.146492141244545774319734e+00, 1},
 	fi{8.003414490659126375852113e+00, 1},
@@ -736,6 +748,21 @@ var frexpSC = []fi{
 	fi{NaN(), 0},
 }
 
+var vfgammaSC = []float64{
+	Inf(-1),
+	-3,
+	0,
+	Inf(1),
+	NaN(),
+}
+var gammaSC = []float64{
+	Inf(-1),
+	Inf(1),
+	Inf(1),
+	Inf(1),
+	NaN(),
+}
+
 var vfhypotSC = [][2]float64{
 	[2]float64{Inf(-1), Inf(-1)},
 	[2]float64{Inf(-1), 0},
@@ -1278,6 +1305,19 @@ func TestFrexp(t *testing.T) {
 	}
 }
 
+func TestGamma(t *testing.T) {
+	for i := 0; i < len(vf); i++ {
+		if f := Gamma(vf[i]); !close(gamma[i], f) {
+			t.Errorf("Gamma(%g) = %g, want %g\n", vf[i], f, gamma[i])
+		}
+	}
+	for i := 0; i < len(vfgammaSC); i++ {
+		if f := Gamma(vfgammaSC[i]); !alike(gammaSC[i], f) {
+			t.Errorf("Gamma(%g) = %g, want %g\n", vfgammaSC[i], f, gammaSC[i])
+		}
+	}
+}
+
 func TestHypot(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
 		a := Fabs(1e200 * tanh[i] * Sqrt(2))
@@ -1748,6 +1788,12 @@ func BenchmarkFrexp(b *testing.B) {
 	}
 }
 
+func BenchmarkGamma(b *testing.B) {
+	for i := 0; i < b.N; i++ {
+		Gamma(2.5)
+	}
+}
+
 func BenchmarkHypot(b *testing.B) {
 	for i := 0; i < b.N; i++ {
 		Hypot(3, 4)

src/pkg/math/gamma.go

+// 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
+
+// The original C code, the long comment, and the constants
+// below are from http://netlib.sandia.gov/cephes/cprob/gamma.c.
+// The go code is a simplified version of the original C.
+//
+//      tgamma.c
+//
+//      Gamma function
+//
+// SYNOPSIS:
+//
+// double x, y, tgamma();
+// extern int signgam;
+//
+// y = tgamma( x );
+//
+// DESCRIPTION:
+//
+// Returns gamma function of the argument.  The result is
+// correctly signed, and the sign (+1 or -1) is also
+// returned in a global (extern) variable named signgam.
+// This variable is also filled in by the logarithmic gamma
+// function lgamma().
+//
+// Arguments |x| <= 34 are reduced by recurrence and the function
+// approximated by a rational function of degree 6/7 in the
+// interval (2,3).  Large arguments are handled by Stirling's
+// formula. Large negative arguments are made positive using
+// a reflection formula.
+//
+// ACCURACY:
+//
+//                      Relative error:
+// arithmetic   domain     # trials      peak         rms
+//    DEC      -34, 34      10000       1.3e-16     2.5e-17
+//    IEEE    -170,-33      20000       2.3e-15     3.3e-16
+//    IEEE     -33,  33     20000       9.4e-16     2.2e-16
+//    IEEE      33, 171.6   20000       2.3e-15     3.2e-16
+//
+// Error for arguments outside the test range will be larger
+// owing to error amplification by the exponential function.
+//
+// Cephes Math Library Release 2.8:  June, 2000
+// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
+//
+// The readme file at http://netlib.sandia.gov/cephes/ says:
+//    Some software in this archive may be from the book _Methods and
+// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
+// International, 1989) or from the Cephes Mathematical Library, a
+// commercial product. In either event, it is copyrighted by the author.
+// What you see here may be used freely but it comes with no support or
+// guarantee.
+//
+//   The two known misprints in the book are repaired here in the
+// source listings for the gamma function and the incomplete beta
+// integral.
+//
+//   Stephen L. Moshier
+//   moshier@na-net.ornl.gov
+
+var _P = []float64{
+	1.60119522476751861407e-04,
+	1.19135147006586384913e-03,
+	1.04213797561761569935e-02,
+	4.76367800457137231464e-02,
+	2.07448227648435975150e-01,
+	4.94214826801497100753e-01,
+	9.99999999999999996796e-01,
+}
+var _Q = []float64{
+	-2.31581873324120129819e-05,
+	5.39605580493303397842e-04,
+	-4.45641913851797240494e-03,
+	1.18139785222060435552e-02,
+	3.58236398605498653373e-02,
+	-2.34591795718243348568e-01,
+	7.14304917030273074085e-02,
+	1.00000000000000000320e+00,
+}
+var _S = []float64{
+	7.87311395793093628397e-04,
+	-2.29549961613378126380e-04,
+	-2.68132617805781232825e-03,
+	3.47222221605458667310e-03,
+	8.33333333333482257126e-02,
+}
+
+// Gamma function computed by Stirling's formula.
+// The polynomial is valid for 33 <= x <= 172.
+func stirling(x float64) float64 {
+	const (
+		SqrtTwoPi   = 2.506628274631000502417
+		MaxStirling = 143.01608
+	)
+	w := 1 / x
+	w = 1 + w*((((_S[0]*w+_S[1])*w+_S[2])*w+_S[3])*w+_S[4])
+	y := Exp(x)
+	if x > MaxStirling { // avoid Pow() overflow
+		v := Pow(x, 0.5*x-0.25)
+		y = v * (v / y)
+	} else {
+		y = Pow(x, x-0.5) / y
+	}
+	y = SqrtTwoPi * y * w
+	return y
+}
+
+// Gamma(x) returns the Gamma function of x.
+//
+// Special cases are:
+//	Gamma(Inf) = Inf
+//	Gamma(-Inf) = -Inf
+//	Gamma(NaN) = NaN
+// Large values overflow to +Inf.
+// Negative integer values equal ±Inf.
+func Gamma(x float64) float64 {
+	const Euler = 0.57721566490153286060651209008240243104215933593992 // A001620
+	// special cases
+	switch {
+	case x < -MaxFloat64 || x != x: // IsInf(x, -1) || IsNaN(x):
+		return x
+	case x < -170.5674972726612 || x > 171.61447887182298:
+		return Inf(1)
+	}
+	q := Fabs(x)
+	p := Floor(q)
+	if q > 33 {
+		if x >= 0 {
+			return stirling(x)
+		}
+		signgam := 1
+		if ip := int(p); ip&1 == 0 {
+			signgam = -1
+		}
+		z := q - p
+		if z > 0.5 {
+			p = p + 1
+			z = q - p
+		}
+		z = q * Sin(Pi*z)
+		if z == 0 {
+			return Inf(signgam)
+		}
+		z = Pi / (Fabs(z) * stirling(q))
+		return float64(signgam) * z
+	}
+
+	// Reduce argument
+	z := float64(1)
+	for x >= 3 {
+		x = x - 1
+		z = z * x
+	}
+	for x < 0 {
+		if x > -1e-09 {
+			goto small
+		}
+		z = z / x
+		x = x + 1
+	}
+	for x < 2 {
+		if x < 1e-09 {
+			goto small
+		}
+		z = z / x
+		x = x + 1
+	}
+
+	if x == 2 {
+		return z
+	}
+
+	x = x - 2
+	p = (((((x*_P[0]+_P[1])*x+_P[2])*x+_P[3])*x+_P[4])*x+_P[5])*x + _P[6]
+	q = ((((((x*_Q[0]+_Q[1])*x+_Q[2])*x+_Q[3])*x+_Q[4])*x+_Q[5])*x+_Q[6])*x + _Q[7]
+	return z * p / q
+
+small:
+	if x == 0 {
+		return Inf(1)
+	}
+	return z / ((1 + Euler*x) * x)
+}