libgo/go/math/sin.go

   1 // Copyright 2011 The Go Authors. All rights reserved.
   2 // Use of this source code is governed by a BSD-style
   3 // license that can be found in the LICENSE file.
   4
   5 package math
   6
   7 /*
   8         Floating-point sine and cosine.
   9 */
  10
  11 // The original C code, the long comment, and the constants
  12 // below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
  13 // available from http://www.netlib.org/cephes/cmath.tgz.
  14 // The go code is a simplified version of the original C.
  15 //
  16 //      sin.c
  17 //
  18 //      Circular sine
  19 //
  20 // SYNOPSIS:
  21 //
  22 // double x, y, sin();
  23 // y = sin( x );
  24 //
  25 // DESCRIPTION:
  26 //
  27 // Range reduction is into intervals of pi/4.  The reduction error is nearly
  28 // eliminated by contriving an extended precision modular arithmetic.
  29 //
  30 // Two polynomial approximating functions are employed.
  31 // Between 0 and pi/4 the sine is approximated by
  32 //      x  +  x**3 P(x**2).
  33 // Between pi/4 and pi/2 the cosine is represented as
  34 //      1  -  x**2 Q(x**2).
  35 //
  36 // ACCURACY:
  37 //
  38 //                      Relative error:
  39 // arithmetic   domain      # trials      peak         rms
  40 //    DEC       0, 10       150000       3.0e-17     7.8e-18
  41 //    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-17
  42 //
  43 // Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9.  The loss
  44 // is not gradual, but jumps suddenly to about 1 part in 10e7.  Results may
  45 // be meaningless for x > 2**49 = 5.6e14.
  46 //
  47 //      cos.c
  48 //
  49 //      Circular cosine
  50 //
  51 // SYNOPSIS:
  52 //
  53 // double x, y, cos();
  54 // y = cos( x );
  55 //
  56 // DESCRIPTION:
  57 //
  58 // Range reduction is into intervals of pi/4.  The reduction error is nearly
  59 // eliminated by contriving an extended precision modular arithmetic.
  60 //
  61 // Two polynomial approximating functions are employed.
  62 // Between 0 and pi/4 the cosine is approximated by
  63 //      1  -  x**2 Q(x**2).
  64 // Between pi/4 and pi/2 the sine is represented as
  65 //      x  +  x**3 P(x**2).
  66 //
  67 // ACCURACY:
  68 //
  69 //                      Relative error:
  70 // arithmetic   domain      # trials      peak         rms
  71 //    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-17
  72 //    DEC        0,+1.07e9   17000       3.0e-17     7.2e-18
  73 //
  74 // Cephes Math Library Release 2.8:  June, 2000
  75 // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
  76 //
  77 // The readme file at http://netlib.sandia.gov/cephes/ says:
  78 //    Some software in this archive may be from the book _Methods and
  79 // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
  80 // International, 1989) or from the Cephes Mathematical Library, a
  81 // commercial product. In either event, it is copyrighted by the author.
  82 // What you see here may be used freely but it comes with no support or
  83 // guarantee.
  84 //
  85 //   The two known misprints in the book are repaired here in the
  86 // source listings for the gamma function and the incomplete beta
  87 // integral.
  88 //
  89 //   Stephen L. Moshier
  90 //   moshier@na-net.ornl.gov
  91
  92 // sin coefficients
  93 var _sin = [...]float64{
  94         1.58962301576546568060E-10, // 0x3de5d8fd1fd19ccd
  95         -2.50507477628578072866E-8, // 0xbe5ae5e5a9291f5d
  96         2.75573136213857245213E-6,  // 0x3ec71de3567d48a1
  97         -1.98412698295895385996E-4, // 0xbf2a01a019bfdf03
  98         8.33333333332211858878E-3,  // 0x3f8111111110f7d0
  99         -1.66666666666666307295E-1, // 0xbfc5555555555548
 100 }
 101
 102 // cos coefficients
 103 var _cos = [...]float64{
 104         -1.13585365213876817300E-11, // 0xbda8fa49a0861a9b
 105         2.08757008419747316778E-9,   // 0x3e21ee9d7b4e3f05
 106         -2.75573141792967388112E-7,  // 0xbe927e4f7eac4bc6
 107         2.48015872888517045348E-5,   // 0x3efa01a019c844f5
 108         -1.38888888888730564116E-3,  // 0xbf56c16c16c14f91
 109         4.16666666666665929218E-2,   // 0x3fa555555555554b
 110 }
 111
 112 // Cos returns the cosine of the radian argument x.
 113 //
 114 // Special cases are:
 115 //      Cos(±Inf) = NaN
 116 //      Cos(NaN) = NaN
 117
 118 //extern cos
 119 func libc_cos(float64) float64
 120
 121 func Cos(x float64) float64 {
 122         return libc_cos(x)
 123 }
 124
 125 func cos(x float64) float64 {
 126         const (
 127                 PI4A = 7.85398125648498535156E-1                             // 0x3fe921fb40000000, Pi/4 split into three parts
 128                 PI4B = 3.77489470793079817668E-8                             // 0x3e64442d00000000,
 129                 PI4C = 2.69515142907905952645E-15                            // 0x3ce8469898cc5170,
 130                 M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
 131         )
 132         // special cases
 133         switch {
 134         case IsNaN(x) || IsInf(x, 0):
 135                 return NaN()
 136         }
 137
 138         // make argument positive
 139         sign := false
 140         if x < 0 {
 141                 x = -x
 142         }
 143
 144         j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
 145         y := float64(j)      // integer part of x/(Pi/4), as float
 146
 147         // map zeros to origin
 148         if j&1 == 1 {
 149                 j++
 150                 y++
 151         }
 152         j &= 7 // octant modulo 2Pi radians (360 degrees)
 153         if j > 3 {
 154                 j -= 4
 155                 sign = !sign
 156         }
 157         if j > 1 {
 158                 sign = !sign
 159         }
 160
 161         z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
 162         zz := z * z
 163         if j == 1 || j == 2 {
 164                 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
 165         } else {
 166                 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])
 167         }
 168         if sign {
 169                 y = -y
 170         }
 171         return y
 172 }
 173
 174 // Sin returns the sine of the radian argument x.
 175 //
 176 // Special cases are:
 177 //      Sin(±0) = ±0
 178 //      Sin(±Inf) = NaN
 179 //      Sin(NaN) = NaN
 180
 181 //extern sin
 182 func libc_sin(float64) float64
 183
 184 func Sin(x float64) float64 {
 185         return libc_sin(x)
 186 }
 187
 188 func sin(x float64) float64 {
 189         const (
 190                 PI4A = 7.85398125648498535156E-1                             // 0x3fe921fb40000000, Pi/4 split into three parts
 191                 PI4B = 3.77489470793079817668E-8                             // 0x3e64442d00000000,
 192                 PI4C = 2.69515142907905952645E-15                            // 0x3ce8469898cc5170,
 193                 M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
 194         )
 195         // special cases
 196         switch {
 197         case x == 0 || IsNaN(x):
 198                 return x // return ±0 || NaN()
 199         case IsInf(x, 0):
 200                 return NaN()
 201         }
 202
 203         // make argument positive but save the sign
 204         sign := false
 205         if x < 0 {
 206                 x = -x
 207                 sign = true
 208         }
 209
 210         j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
 211         y := float64(j)      // integer part of x/(Pi/4), as float
 212
 213         // map zeros to origin
 214         if j&1 == 1 {
 215                 j++
 216                 y++
 217         }
 218         j &= 7 // octant modulo 2Pi radians (360 degrees)
 219         // reflect in x axis
 220         if j > 3 {
 221                 sign = !sign
 222                 j -= 4
 223         }
 224
 225         z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
 226         zz := z * z
 227         if j == 1 || j == 2 {
 228                 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])
 229         } else {
 230                 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
 231         }
 232         if sign {
 233                 y = -y
 234         }
 235         return y
 236 }