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 5package 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 93var _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 103var _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 119func libc_cos(float64) float64 120 121func Cos(x float64) float64 { 122 return libc_cos(x) 123} 124 125func 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 186func libc_sin(float64) float64 187 188func Sin(x float64) float64 { 189 return libc_sin(x) 190} 191 192func 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} 245