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 (a] 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 func Cos(x float64) float64 118 119 func cos(x float64) float64 { 120 const ( 121 PI4A = 7.85398125648498535156E-1 // 0x3fe921fb40000000, Pi/4 split into three parts 122 PI4B = 3.77489470793079817668E-8 // 0x3e64442d00000000, 123 PI4C = 2.69515142907905952645E-15 // 0x3ce8469898cc5170, 124 M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi 125 ) 126 // special cases 127 switch { 128 case IsNaN(x) || IsInf(x, 0): 129 return NaN() 130 } 131 132 // make argument positive 133 sign := false 134 if x < 0 { 135 x = -x 136 } 137 138 j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle 139 y := float64(j) // integer part of x/(Pi/4), as float 140 141 // map zeros to origin 142 if j&1 == 1 { 143 j++ 144 y++ 145 } 146 j &= 7 // octant modulo 2Pi radians (360 degrees) 147 if j > 3 { 148 j -= 4 149 sign = !sign 150 } 151 if j > 1 { 152 sign = !sign 153 } 154 155 z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic 156 zz := z * z 157 if j == 1 || j == 2 { 158 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5]) 159 } else { 160 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]) 161 } 162 if sign { 163 y = -y 164 } 165 return y 166 } 167 168 // Sin returns the sine of the radian argument x. 169 // 170 // Special cases are: 171 // Sin(0) = 0 172 // Sin(Inf) = NaN 173 // Sin(NaN) = NaN 174 func Sin(x float64) float64 175 176 func sin(x float64) float64 { 177 const ( 178 PI4A = 7.85398125648498535156E-1 // 0x3fe921fb40000000, Pi/4 split into three parts 179 PI4B = 3.77489470793079817668E-8 // 0x3e64442d00000000, 180 PI4C = 2.69515142907905952645E-15 // 0x3ce8469898cc5170, 181 M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi 182 ) 183 // special cases 184 switch { 185 case x == 0 || IsNaN(x): 186 return x // return 0 || NaN() 187 case IsInf(x, 0): 188 return NaN() 189 } 190 191 // make argument positive but save the sign 192 sign := false 193 if x < 0 { 194 x = -x 195 sign = true 196 } 197 198 j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle 199 y := float64(j) // integer part of x/(Pi/4), as float 200 201 // map zeros to origin 202 if j&1 == 1 { 203 j++ 204 y++ 205 } 206 j &= 7 // octant modulo 2Pi radians (360 degrees) 207 // reflect in x axis 208 if j > 3 { 209 sign = !sign 210 j -= 4 211 } 212 213 z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic 214 zz := z * z 215 if j == 1 || j == 2 { 216 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]) 217 } else { 218 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5]) 219 } 220 if sign { 221 y = -y 222 } 223 return y 224 } 225
