math: Improved accuracy for Sin and Cos.

Fixes #1564.

R=rsc, dchest
CC=golang-dev
https://golang.org/cl/5320056

src/pkg/math/all_test.go

@@ -1709,7 +1709,7 @@ func TestCopysign(t *testing.T) {
 
 func TestCos(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if f := Cos(vf[i]); !close(cos[i], f) {
+		if f := Cos(vf[i]); !veryclose(cos[i], f) {
 			t.Errorf("Cos(%g) = %g, want %g", vf[i], f, cos[i])
 		}
 	}
@@ -2192,7 +2192,7 @@ func TestSignbit(t *testing.T) {
 }
 func TestSin(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if f := Sin(vf[i]); !close(sin[i], f) {
+		if f := Sin(vf[i]); !veryclose(sin[i], f) {
 			t.Errorf("Sin(%g) = %g, want %g", vf[i], f, sin[i])
 		}
 	}
@@ -2205,7 +2205,7 @@ func TestSin(t *testing.T) {
 
 func TestSincos(t *testing.T) {
 	for i := 0; i < len(vf); i++ {
-		if s, c := Sincos(vf[i]); !close(sin[i], s) || !close(cos[i], c) {
+		if s, c := Sincos(vf[i]); !veryclose(sin[i], s) || !veryclose(cos[i], c) {
 			t.Errorf("Sincos(%g) = %g, %g want %g, %g", vf[i], s, c, sin[i], cos[i])
 		}
 	}

src/pkg/math/sin.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.
 
@@ -6,60 +6,218 @@ package math
 
 /*
 	Floating-point sine and cosine.
-
-	Coefficients are #5077 from Hart & Cheney. (18.80D)
 */
 
-func sinus(x float64, quad int) float64 {
+// 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.
+//
+//      sin.c
+//
+//      Circular sine
+//
+// SYNOPSIS:
+//
+// double x, y, sin();
+// y = sin( x );
+//
+// DESCRIPTION:
+//
+// Range reduction is into intervals of pi/4.  The reduction error is nearly
+// eliminated by contriving an extended precision modular arithmetic.
+//
+// Two polynomial approximating functions are employed.
+// Between 0 and pi/4 the sine is approximated by
+//      x  +  x**3 P(x**2).
+// Between pi/4 and pi/2 the cosine is represented as
+//      1  -  x**2 Q(x**2).
+//
+// ACCURACY:
+//
+//                      Relative error:
+// arithmetic   domain      # trials      peak         rms
+//    DEC       0, 10       150000       3.0e-17     7.8e-18
+//    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-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.
+//
+//      cos.c
+//
+//      Circular cosine
+//
+// SYNOPSIS:
+//
+// double x, y, cos();
+// y = cos( x );
+//
+// DESCRIPTION:
+//
+// Range reduction is into intervals of pi/4.  The reduction error is nearly
+// eliminated by contriving an extended precision modular arithmetic.
+//
+// Two polynomial approximating functions are employed.
+// Between 0 and pi/4 the cosine is approximated by
+//      1  -  x**2 Q(x**2).
+// Between pi/4 and pi/2 the sine is represented as
+//      x  +  x**3 P(x**2).
+//
+// ACCURACY:
+//
+//                      Relative error:
+// arithmetic   domain      # trials      peak         rms
+//    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-17
+//    DEC        0,+1.07e9   17000       3.0e-17     7.2e-18
+//
+// 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
+
+// sin coefficients
+var _sin = [...]float64{
+	1.58962301576546568060E-10, // 0x3de5d8fd1fd19ccd
+	-2.50507477628578072866E-8, // 0xbe5ae5e5a9291f5d
+	2.75573136213857245213E-6,  // 0x3ec71de3567d48a1
+	-1.98412698295895385996E-4, // 0xbf2a01a019bfdf03
+	8.33333333332211858878E-3,  // 0x3f8111111110f7d0
+	-1.66666666666666307295E-1, // 0xbfc5555555555548
+}
+// cos coefficients
+var _cos = [...]float64{
+	-1.13585365213876817300E-11, // 0xbda8fa49a0861a9b
+	2.08757008419747316778E-9,   // 0x3e21ee9d7b4e3f05
+	-2.75573141792967388112E-7,  // 0xbe927e4f7eac4bc6
+	2.48015872888517045348E-5,   // 0x3efa01a019c844f5
+	-1.38888888888730564116E-3,  // 0xbf56c16c16c14f91
+	4.16666666666665929218E-2,   // 0x3fa555555555554b
+}
+
+// Cos returns the cosine of x.
+//
+// Special conditions are:
+//	Cos(±Inf) = NaN
+//	Cos(NaN) = NaN
+func Cos(x float64) float64 {
 	const (
-		P0 = .1357884097877375669092680e8
-		P1 = -.4942908100902844161158627e7
-		P2 = .4401030535375266501944918e6
-		P3 = -.1384727249982452873054457e5
-		P4 = .1459688406665768722226959e3
-		Q0 = .8644558652922534429915149e7
-		Q1 = .4081792252343299749395779e6
-		Q2 = .9463096101538208180571257e4
-		Q3 = .1326534908786136358911494e3
+		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 != x || x < -MaxFloat64 || x > MaxFloat64: // IsNaN(x) || IsInf(x, 0):
+		return NaN()
+	}
+
+	// make argument positive
+	sign := false
 	if x < 0 {
 		x = -x
-		quad = quad + 2
 	}
-	x = x * (2 / Pi) /* underflow? */
-	var y float64
-	if x > 32764 {
-		var e float64
-		e, y = Modf(x)
-		e = e + float64(quad)
-		f, _ := Modf(0.25 * e)
-		quad = int(e - 4*f)
-	} else {
-		k := int32(x)
-		y = x - float64(k)
-		quad = (quad + int(k)) & 3
+
+	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
+
+	// map zeros to origin
+	if j&1 == 1 {
+		j += 1
+		y += 1
+	}
+	j &= 7 // octant modulo 2Pi radians (360 degrees)
+	if j > 3 {
+		j -= 4
+		sign = !sign
+	}
+	if j > 1 {
+		sign = !sign
 	}
 
-	if quad&1 != 0 {
-		y = 1 - y
+	z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
+	zz := z * z
+	if j == 1 || j == 2 {
+		y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
+	} else {
+		y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
 	}
-	if quad > 1 {
+	if sign {
 		y = -y
 	}
-
-	yy := y * y
-	temp1 := ((((P4*yy+P3)*yy+P2)*yy+P1)*yy + P0) * y
-	temp2 := ((((yy+Q3)*yy+Q2)*yy+Q1)*yy + Q0)
-	return temp1 / temp2
+	return y
 }
 
-// Cos returns the cosine of x.
-func Cos(x float64) float64 {
+// Sin returns the sine of x.
+//
+// Special conditions are:
+//	Sin(±0) = ±0
+//	Sin(±Inf) = NaN
+//	Sin(NaN) = NaN
+func Sin(x float64) float64 {
+	const (
+		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()
+	}
+
+	// make argument positive but save the sign
+	sign := false
 	if x < 0 {
 		x = -x
+		sign = true
 	}
-	return sinus(x, 1)
-}
 
-// Sin returns the sine of x.
-func Sin(x float64) float64 { return sinus(x, 0) }
+	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
+
+	// map zeros to origin
+	if j&1 == 1 {
+		j += 1
+		y += 1
+	}
+	j &= 7 // octant modulo 2Pi radians (360 degrees)
+	// reflect in x axis
+	if j > 3 {
+		sign = !sign
+		j -= 4
+	}
+
+	z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
+	zz := z * z
+	if j == 1 || j == 2 {
+		y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
+	} else {
+		y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
+	}
+	if sign {
+		y = -y
+	}
+	return y
+}