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 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 ) 125 // special cases 126 switch { 127 case IsNaN(x) || IsInf(x, 0): 128 return NaN() 129 } 130 131 // make argument positive 132 sign := false 133 x = Abs(x) 134 135 var j uint64 136 var y, z float64 137 if x >= reduceThreshold { 138 j, z = trigReduce(x) 139 } else { 140 j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle 141 y = float64(j) // integer part of x/(Pi/4), as float 142 143 // map zeros to origin 144 if j&1 == 1 { 145 j++ 146 y++ 147 } 148 j &= 7 // octant modulo 2Pi radians (360 degrees) 149 z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic 150 } 151 152 if j > 3 { 153 j -= 4 154 sign = !sign 155 } 156 if j > 1 { 157 sign = !sign 158 } 159 160 zz := z * z 161 if j == 1 || j == 2 { 162 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5]) 163 } else { 164 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]) 165 } 166 if sign { 167 y = -y 168 } 169 return y 170 } 171 172 // Sin returns the sine of the radian argument x. 173 // 174 // Special cases are: 175 // Sin(±0) = ±0 176 // Sin(±Inf) = NaN 177 // Sin(NaN) = NaN 178 func Sin(x float64) float64 179 180 func sin(x float64) float64 { 181 const ( 182 PI4A = 7.85398125648498535156e-1 // 0x3fe921fb40000000, Pi/4 split into three parts 183 PI4B = 3.77489470793079817668e-8 // 0x3e64442d00000000, 184 PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170, 185 ) 186 // special cases 187 switch { 188 case x == 0 || IsNaN(x): 189 return x // return ±0 || NaN() 190 case IsInf(x, 0): 191 return NaN() 192 } 193 194 // make argument positive but save the sign 195 sign := false 196 if x < 0 { 197 x = -x 198 sign = true 199 } 200 201 var j uint64 202 var y, z float64 203 if x >= reduceThreshold { 204 j, z = trigReduce(x) 205 } else { 206 j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle 207 y = float64(j) // integer part of x/(Pi/4), as float 208 209 // map zeros to origin 210 if j&1 == 1 { 211 j++ 212 y++ 213 } 214 j &= 7 // octant modulo 2Pi radians (360 degrees) 215 z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic 216 } 217 // reflect in x axis 218 if j > 3 { 219 sign = !sign 220 j -= 4 221 } 222 zz := z * z 223 if j == 1 || j == 2 { 224 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]) 225 } else { 226 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5]) 227 } 228 if sign { 229 y = -y 230 } 231 return y 232 } 233