math: Faster Tanh

From 159 to 47.6 ns/op; slightly more accurate.

R=rsc, golang-dev, mtj, dave, remyoudompheng
CC=golang-dev
https://golang.org/cl/6500121

src/pkg/math/tanh.go

@@ -4,12 +4,66 @@
 
 package math
 
-/*
-	Floating-point hyperbolic tangent.
+// The original C code, the long comment, and the constants
+// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
+// available from http://www.netlib.org/cephes/cmath.tgz.
+// The go code is a simplified version of the original C.
+//      tanh.c
+//
+//      Hyperbolic tangent
+//
+// SYNOPSIS:
+//
+// double x, y, tanh();
+//
+// y = tanh( x );
+//
+// DESCRIPTION:
+//
+// Returns hyperbolic tangent of argument in the range MINLOG to MAXLOG.
+//      MAXLOG = 8.8029691931113054295988e+01 = log(2**127)
+//      MINLOG = -8.872283911167299960540e+01 = log(2**-128)
+//
+// A rational function is used for |x| < 0.625.  The form
+// x + x**3 P(x)/Q(x) of Cody & Waite is employed.
+// Otherwise,
+//      tanh(x) = sinh(x)/cosh(x) = 1  -  2/(exp(2x) + 1).
+//
+// ACCURACY:
+//
+//                      Relative error:
+// arithmetic   domain     # trials      peak         rms
+//    IEEE      -2,2        30000       2.5e-16     5.8e-17
+//
+// 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
+//
 
-	Sinh and Cosh are called except for large arguments, which
-	would cause overflow improperly.
-*/
+var tanhP = [...]float64{
+	-9.64399179425052238628E-1,
+	-9.92877231001918586564E1,
+	-1.61468768441708447952E3,
+}
+var tanhQ = [...]float64{
+	1.12811678491632931402E2,
+	2.23548839060100448583E3,
+	4.84406305325125486048E3,
+}
 
 // Tanh computes the hyperbolic tangent of x.
 //
@@ -18,15 +72,26 @@ package math
 //	Tanh(±Inf) = ±1
 //	Tanh(NaN) = NaN
 func Tanh(x float64) float64 {
+	const MAXLOG = 8.8029691931113054295988e+01 // log(2**127)
+	z := Abs(x)
+	switch {
+	case z > 0.5*MAXLOG:
 		if x < 0 {
-		x = -x
-		if x > 21 {
 			return -1
 		}
-		return -Sinh(x) / Cosh(x)
-	}
-	if x > 21 {
 		return 1
+	case z >= 0.625:
+		s := Exp(2 * z)
+		z = 1 - 2/(s+1)
+		if x < 0 {
+			z = -z
+		}
+	default:
+		if x == 0 {
+			return x
+		}
+		s := x * x
+		z = x + x*s*((tanhP[0]*s+tanhP[1])*s+tanhP[2])/(((s+tanhQ[0])*s+tanhQ[1])*s+tanhQ[2])
 	}
-	return Sinh(x) / Cosh(x)
+	return z
 }