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

src/pkg/math/Makefile

@@ -52,6 +52,7 @@ ALLGOFILES=\
 	fmod.go\
 	frexp.go\
 	hypot.go\
+	lgamma.go\
 	ldexp.go\
 	log.go\
 	log1p.go\

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

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)

src/pkg/math/lgamma.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
+
+/*
+	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
+}

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