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 // [email protected] 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 if haveArchCos { 119 return archCos(x) 120 } 121 return cos(x) 122 } 123 124 func cos(x float64) float64 { 125 const ( 126 PI4A = 7.85398125648498535156e-1 // 0x3fe921fb40000000, Pi/4 split into three parts 127 PI4B = 3.77489470793079817668e-8 // 0x3e64442d00000000, 128 PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170, 129 ) 130 // special cases 131 switch { 132 case IsNaN(x) || IsInf(x, 0): 133 return NaN() 134 } 135 136 // make argument positive 137 sign := false 138 x = Abs(x) 139 140 var j uint64 141 var y, z float64 142 if x >= reduceThreshold { 143 j, z = trigReduce(x) 144 } else { 145 j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle 146 y = float64(j) // integer part of x/(Pi/4), as float 147 148 // map zeros to origin 149 if j&1 == 1 { 150 j++ 151 y++ 152 } 153 j &= 7 // octant modulo 2Pi radians (360 degrees) 154 z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic 155 } 156 157 if j > 3 { 158 j -= 4 159 sign = !sign 160 } 161 if j > 1 { 162 sign = !sign 163 } 164 165 zz := z * z 166 if j == 1 || j == 2 { 167 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5]) 168 } else { 169 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]) 170 } 171 if sign { 172 y = -y 173 } 174 return y 175 } 176 177 // Sin returns the sine of the radian argument x. 178 // 179 // Special cases are: 180 // Sin(±0) = ±0 181 // Sin(±Inf) = NaN 182 // Sin(NaN) = NaN 183 func Sin(x float64) float64 { 184 if haveArchSin { 185 return archSin(x) 186 } 187 return sin(x) 188 } 189 190 func sin(x float64) float64 { 191 const ( 192 PI4A = 7.85398125648498535156e-1 // 0x3fe921fb40000000, Pi/4 split into three parts 193 PI4B = 3.77489470793079817668e-8 // 0x3e64442d00000000, 194 PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170, 195 ) 196 // special cases 197 switch { 198 case x == 0 || IsNaN(x): 199 return x // return ±0 || NaN() 200 case IsInf(x, 0): 201 return NaN() 202 } 203 204 // make argument positive but save the sign 205 sign := false 206 if x < 0 { 207 x = -x 208 sign = true 209 } 210 211 var j uint64 212 var y, z float64 213 if x >= reduceThreshold { 214 j, z = trigReduce(x) 215 } else { 216 j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle 217 y = float64(j) // integer part of x/(Pi/4), as float 218 219 // map zeros to origin 220 if j&1 == 1 { 221 j++ 222 y++ 223 } 224 j &= 7 // octant modulo 2Pi radians (360 degrees) 225 z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic 226 } 227 // reflect in x axis 228 if j > 3 { 229 sign = !sign 230 j -= 4 231 } 232 zz := z * z 233 if j == 1 || j == 2 { 234 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]) 235 } else { 236 y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5]) 237 } 238 if sign { 239 y = -y 240 } 241 return y 242 } 243