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Source file src/math/cmplx/tan.go

Documentation: math/cmplx

     1  // Copyright 2010 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 cmplx
     6  
     7  import "math"
     8  
     9  // The original C code, the long comment, and the constants
    10  // below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
    11  // The go code is a simplified version of the original C.
    12  //
    13  // Cephes Math Library Release 2.8:  June, 2000
    14  // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
    15  //
    16  // The readme file at http://netlib.sandia.gov/cephes/ says:
    17  //    Some software in this archive may be from the book _Methods and
    18  // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
    19  // International, 1989) or from the Cephes Mathematical Library, a
    20  // commercial product. In either event, it is copyrighted by the author.
    21  // What you see here may be used freely but it comes with no support or
    22  // guarantee.
    23  //
    24  //   The two known misprints in the book are repaired here in the
    25  // source listings for the gamma function and the incomplete beta
    26  // integral.
    27  //
    28  //   Stephen L. Moshier
    29  //   moshier@na-net.ornl.gov
    30  
    31  // Complex circular tangent
    32  //
    33  // DESCRIPTION:
    34  //
    35  // If
    36  //     z = x + iy,
    37  //
    38  // then
    39  //
    40  //           sin 2x  +  i sinh 2y
    41  //     w  =  --------------------.
    42  //            cos 2x  +  cosh 2y
    43  //
    44  // On the real axis the denominator is zero at odd multiples
    45  // of PI/2.  The denominator is evaluated by its Taylor
    46  // series near these points.
    47  //
    48  // ctan(z) = -i ctanh(iz).
    49  //
    50  // ACCURACY:
    51  //
    52  //                      Relative error:
    53  // arithmetic   domain     # trials      peak         rms
    54  //    DEC       -10,+10      5200       7.1e-17     1.6e-17
    55  //    IEEE      -10,+10     30000       7.2e-16     1.2e-16
    56  // Also tested by ctan * ccot = 1 and catan(ctan(z))  =  z.
    57  
    58  // Tan returns the tangent of x.
    59  func Tan(x complex128) complex128 {
    60  	d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
    61  	if math.Abs(d) < 0.25 {
    62  		d = tanSeries(x)
    63  	}
    64  	if d == 0 {
    65  		return Inf()
    66  	}
    67  	return complex(math.Sin(2*real(x))/d, math.Sinh(2*imag(x))/d)
    68  }
    69  
    70  // Complex hyperbolic tangent
    71  //
    72  // DESCRIPTION:
    73  //
    74  // tanh z = (sinh 2x  +  i sin 2y) / (cosh 2x + cos 2y) .
    75  //
    76  // ACCURACY:
    77  //
    78  //                      Relative error:
    79  // arithmetic   domain     # trials      peak         rms
    80  //    IEEE      -10,+10     30000       1.7e-14     2.4e-16
    81  
    82  // Tanh returns the hyperbolic tangent of x.
    83  func Tanh(x complex128) complex128 {
    84  	d := math.Cosh(2*real(x)) + math.Cos(2*imag(x))
    85  	if d == 0 {
    86  		return Inf()
    87  	}
    88  	return complex(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d)
    89  }
    90  
    91  // Program to subtract nearest integer multiple of PI
    92  func reducePi(x float64) float64 {
    93  	const (
    94  		// extended precision value of PI:
    95  		DP1 = 3.14159265160560607910E0   // ?? 0x400921fb54000000
    96  		DP2 = 1.98418714791870343106E-9  // ?? 0x3e210b4610000000
    97  		DP3 = 1.14423774522196636802E-17 // ?? 0x3c6a62633145c06e
    98  	)
    99  	t := x / math.Pi
   100  	if t >= 0 {
   101  		t += 0.5
   102  	} else {
   103  		t -= 0.5
   104  	}
   105  	t = float64(int64(t)) // int64(t) = the multiple
   106  	return ((x - t*DP1) - t*DP2) - t*DP3
   107  }
   108  
   109  // Taylor series expansion for cosh(2y) - cos(2x)
   110  func tanSeries(z complex128) float64 {
   111  	const MACHEP = 1.0 / (1 << 53)
   112  	x := math.Abs(2 * real(z))
   113  	y := math.Abs(2 * imag(z))
   114  	x = reducePi(x)
   115  	x = x * x
   116  	y = y * y
   117  	x2 := 1.0
   118  	y2 := 1.0
   119  	f := 1.0
   120  	rn := 0.0
   121  	d := 0.0
   122  	for {
   123  		rn++
   124  		f *= rn
   125  		rn++
   126  		f *= rn
   127  		x2 *= x
   128  		y2 *= y
   129  		t := y2 + x2
   130  		t /= f
   131  		d += t
   132  
   133  		rn++
   134  		f *= rn
   135  		rn++
   136  		f *= rn
   137  		x2 *= x
   138  		y2 *= y
   139  		t = y2 - x2
   140  		t /= f
   141  		d += t
   142  		if !(math.Abs(t/d) > MACHEP) {
   143  			// Caution: Use ! and > instead of <= for correct behavior if t/d is NaN.
   144  			// See issue 17577.
   145  			break
   146  		}
   147  	}
   148  	return d
   149  }
   150  
   151  // Complex circular cotangent
   152  //
   153  // DESCRIPTION:
   154  //
   155  // If
   156  //     z = x + iy,
   157  //
   158  // then
   159  //
   160  //           sin 2x  -  i sinh 2y
   161  //     w  =  --------------------.
   162  //            cosh 2y  -  cos 2x
   163  //
   164  // On the real axis, the denominator has zeros at even
   165  // multiples of PI/2.  Near these points it is evaluated
   166  // by a Taylor series.
   167  //
   168  // ACCURACY:
   169  //
   170  //                      Relative error:
   171  // arithmetic   domain     # trials      peak         rms
   172  //    DEC       -10,+10      3000       6.5e-17     1.6e-17
   173  //    IEEE      -10,+10     30000       9.2e-16     1.2e-16
   174  // Also tested by ctan * ccot = 1 + i0.
   175  
   176  // Cot returns the cotangent of x.
   177  func Cot(x complex128) complex128 {
   178  	d := math.Cosh(2*imag(x)) - math.Cos(2*real(x))
   179  	if math.Abs(d) < 0.25 {
   180  		d = tanSeries(x)
   181  	}
   182  	if d == 0 {
   183  		return Inf()
   184  	}
   185  	return complex(math.Sin(2*real(x))/d, -math.Sinh(2*imag(x))/d)
   186  }
   187  

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