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  	)
   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  

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