Source file src/math/sin.go

     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  //
   116  //	Cos(±Inf) = NaN
   117  //	Cos(NaN) = NaN
   118  func Cos(x float64) float64 {
   119  	if haveArchCos {
   120  		return archCos(x)
   121  	}
   122  	return cos(x)
   123  }
   124  
   125  func 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  //
   182  //	Sin(±0) = ±0
   183  //	Sin(±Inf) = NaN
   184  //	Sin(NaN) = NaN
   185  func Sin(x float64) float64 {
   186  	if haveArchSin {
   187  		return archSin(x)
   188  	}
   189  	return sin(x)
   190  }
   191  
   192  func 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  

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