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         )
 131         // special cases
 132         switch {
 133         case IsNaN(x) || IsInf(x, 0):
 134                 return NaN()
 135         }
 136
 137         // make argument positive
 138         sign := false
 139         x = Abs(x)
 140
 141         var j uint64
 142         var y, z float64
 143         if x >= reduceThreshold {
 144                 j, z = trigReduce(x)
 145         } else {
 146                 j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
 147                 y = float64(j)           // integer part of x/(Pi/4), as float
 148
 149                 // map zeros to origin
 150                 if j&1 == 1 {
 151                         j++
 152                         y++
 153                 }
 154                 j &= 7                               // octant modulo 2Pi radians (360 degrees)
 155                 z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
 156         }
 157
 158         if j > 3 {
 159                 j -= 4
 160                 sign = !sign
 161         }
 162         if j > 1 {
 163                 sign = !sign
 164         }
 165
 166         zz := z * z
 167         if j == 1 || j == 2 {
 168                 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
 169         } else {
 170                 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])
 171         }
 172         if sign {
 173                 y = -y
 174         }
 175         return y
 176 }
 177
 178 // Sin returns the sine of the radian argument x.
 179 //
 180 // Special cases are:
 181 //      Sin(±0) = ±0
 182 //      Sin(±Inf) = NaN
 183 //      Sin(NaN) = NaN
 184
 185 //extern sin
 186 func libc_sin(float64) float64
 187
 188 func Sin(x float64) float64 {
 189         return libc_sin(x)
 190 }
 191
 192 func sin(x float64) float64 {
 193         const (
 194                 PI4A = 7.85398125648498535156E-1  // 0x3fe921fb40000000, Pi/4 split into three parts
 195                 PI4B = 3.77489470793079817668E-8  // 0x3e64442d00000000,
 196                 PI4C = 2.69515142907905952645E-15 // 0x3ce8469898cc5170,
 197         )
 198         // special cases
 199         switch {
 200         case x == 0 || IsNaN(x):
 201                 return x // return ±0 || NaN()
 202         case IsInf(x, 0):
 203                 return NaN()
 204         }
 205
 206         // make argument positive but save the sign
 207         sign := false
 208         if x < 0 {
 209                 x = -x
 210                 sign = true
 211         }
 212
 213         var j uint64
 214         var y, z float64
 215         if x >= reduceThreshold {
 216                 j, z = trigReduce(x)
 217         } else {
 218                 j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
 219                 y = float64(j)           // integer part of x/(Pi/4), as float
 220
 221                 // map zeros to origin
 222                 if j&1 == 1 {
 223                         j++
 224                         y++
 225                 }
 226                 j &= 7                               // octant modulo 2Pi radians (360 degrees)
 227                 z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
 228         }
 229         // reflect in x axis
 230         if j > 3 {
 231                 sign = !sign
 232                 j -= 4
 233         }
 234         zz := z * z
 235         if j == 1 || j == 2 {
 236                 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])
 237         } else {
 238                 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
 239         }
 240         if sign {
 241                 y = -y
 242         }
 243         return y
 244 }