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Source file src/math/sin.go

Documentation: math

     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  		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  

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