-// 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.
/*
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
+}