math: improved accuracy for Tan

R=rsc
CC=golang-dev
https://golang.org/cl/5298087

src/pkg/math/all_test.go

@@ -7,7 +7,6 @@ package math_test
 import (
 	"fmt"
 	. "math"
-	"runtime"
 	"testing"
 )
 
@@ -2247,7 +2246,7 @@ func TestSqrt(t *testing.T) {
 
 func TestTan(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if f := Tan(vf[i]); !close(tan[i], f) {
+		if f := Tan(vf[i]); !veryclose(tan[i], f) {
 			t.Errorf("Tan(%g) = %g, want %g", vf[i], f, tan[i])
 		}
 	}
@@ -2257,16 +2256,6 @@ func TestTan(t *testing.T) {
 			t.Errorf("Tan(%g) = %g, want %g", vfsinSC[i], f, sinSC[i])
 		}
 	}
-
-	// Make sure portable Tan(Pi/2) doesn't panic (it used to).
-	// The portable implementation returns NaN.
-	// Assembly implementations might not,
-	// because Pi/2 is not exactly representable.
-	if runtime.GOARCH != "386" {
-		if f := Tan(Pi / 2); !alike(f, NaN()) {
-			t.Errorf("Tan(%g) = %g, want %g", Pi/2, f, NaN())
-		}
-	}
 }
 
 func TestTanh(t *testing.T) {

src/pkg/math/tan.go

-// Copyright 2009 The Go Authors. All rights reserved.
+// Copyright 2011 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 tangent.
+	Floating-point 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.
+//
+//      tan.c
+//
+//      Circular tangent
+//
+// SYNOPSIS:
+//
+// double x, y, tan();
+// y = tan( x );
+//
+// DESCRIPTION:
+//
+// Returns the circular tangent of the radian argument x.
+//
+// Range reduction is modulo pi/4.  A rational function
+//       x + x**3 P(x**2)/Q(x**2)
+// is employed in the basic interval [0, pi/4].
+//
+// ACCURACY:
+//                      Relative error:
+// arithmetic   domain     # trials      peak         rms
+//    DEC      +-1.07e9      44000      4.1e-17     1.0e-17
+//    IEEE     +-1.07e9      30000      2.9e-16     8.1e-17
+//
+// Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9.  The loss
+// is not gradual, but jumps suddenly to about 1 part in 10e7.  Results may
+// be meaningless for x > 2**49 = 5.6e14.
+// [Accuracy loss statement from sin.go comments.]
+//
+// 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
+
+// tan coefficients
+var _tanP = [...]float64{
+	-1.30936939181383777646E4, // 0xc0c992d8d24f3f38
+	1.15351664838587416140E6,  // 0x413199eca5fc9ddd
+	-1.79565251976484877988E7, // 0xc1711fead3299176
+}
+var _tanQ = [...]float64{
+	1.00000000000000000000E0,
+	1.36812963470692954678E4,  //0x40cab8a5eeb36572
+	-1.32089234440210967447E6, //0xc13427bc582abc96
+	2.50083801823357915839E7,  //0x4177d98fc2ead8ef
+	-5.38695755929454629881E7, //0xc189afe03cbe5a31
+}
+
 // Tan returns the tangent of x.
+//
+// Special conditions are:
+//	Tan(±0) = ±0
+//	Tan(±Inf) = NaN
+//	Tan(NaN) = NaN
 func Tan(x float64) float64 {
-	// Coefficients are #4285 from Hart & Cheney. (19.74D)
 	const (
-		P0 = -.1306820264754825668269611177e+5
-		P1 = .1055970901714953193602353981e+4
-		P2 = -.1550685653483266376941705728e+2
-		P3 = .3422554387241003435328470489e-1
-		P4 = .3386638642677172096076369e-4
-		Q0 = -.1663895238947119001851464661e+5
-		Q1 = .4765751362916483698926655581e+4
-		Q2 = -.1555033164031709966900124574e+3
+		PI4A = 7.85398125648498535156E-1                             // 0x3fe921fb40000000, Pi/4 split into three parts
+		PI4B = 3.77489470793079817668E-8                             // 0x3e64442d00000000,
+		PI4C = 2.69515142907905952645E-15                            // 0x3ce8469898cc5170,
+		M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
 	)
+	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
+	// when compiler does it for us
+	// special cases
+	switch {
+	case x == 0 || x != x: // x == 0 || IsNaN():
+		return x // return ±0 || NaN()
+	case x < -MaxFloat64 || x > MaxFloat64: // IsInf(x, 0):
+		return NaN()
+	}
 
-	flag := false
+	// make argument positive but save the sign
 	sign := false
 	if x < 0 {
 		x = -x
 		sign = true
 	}
-	x = x * (4 / Pi) /* overflow? */
-	var e float64
-	e, x = Modf(x)
-	i := int32(e)
-
-	switch i & 3 {
-	case 1:
-		x = 1 - x
-		flag = true
 
-	case 2:
-		sign = !sign
-		flag = true
+	j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
+	y := float64(j)      // integer part of x/(Pi/4), as float
 
-	case 3:
-		x = 1 - x
-		sign = !sign
+	/* map zeros and singularities to origin */
+	if j&1 == 1 {
+		j += 1
+		y += 1
 	}
 
-	xsq := x * x
-	temp := ((((P4*xsq+P3)*xsq+P2)*xsq+P1)*xsq + P0) * x
-	temp = temp / (((xsq+Q2)*xsq+Q1)*xsq + Q0)
+	z := ((x - y*PI4A) - y*PI4B) - y*PI4C
+	zz := z * z
 
-	if flag {
-		if temp == 0 {
-			return NaN()
-		}
-		temp = 1 / temp
+	if zz > 1e-14 {
+		y = z + z*(zz*(((_tanP[0]*zz)+_tanP[1])*zz+_tanP[2])/((((zz+_tanQ[1])*zz+_tanQ[2])*zz+_tanQ[3])*zz+_tanQ[4]))
+	} else {
+		y = z
+	}
+	if j&2 == 2 {
+		y = -1 / y
 	}
 	if sign {
-		temp = -temp
+		y = -y
 	}
-	return temp
+	return y
 }