asin.go

     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 arc sine
    32  //
    33  // DESCRIPTION:
    34  //
    35  // Inverse complex sine:
    36  //                               2
    37  // w = -i clog( iz + csqrt( 1 - z ) ).
    38  //
    39  // casin(z) = -i casinh(iz)
    40  //
    41  // ACCURACY:
    42  //
    43  //                      Relative error:
    44  // arithmetic   domain     # trials      peak         rms
    45  //    DEC       -10,+10     10100       2.1e-15     3.4e-16
    46  //    IEEE      -10,+10     30000       2.2e-14     2.7e-15
    47  // Larger relative error can be observed for z near zero.
    48  // Also tested by csin(casin(z)) = z.
    49  
    50  // Asin returns the inverse sine of x.
    51  func Asin(x complex128) complex128 {
    52  	switch re, im := real(x), imag(x); {
    53  	case im == 0 && math.Abs(re) <= 1:
    54  		return complex(math.Asin(re), im)
    55  	case re == 0 && math.Abs(im) <= 1:
    56  		return complex(re, math.Asinh(im))
    57  	case math.IsNaN(im):
    58  		switch {
    59  		case re == 0:
    60  			return complex(re, math.NaN())
    61  		case math.IsInf(re, 0):
    62  			return complex(math.NaN(), re)
    63  		default:
    64  			return NaN()
    65  		}
    66  	case math.IsInf(im, 0):
    67  		switch {
    68  		case math.IsNaN(re):
    69  			return x
    70  		case math.IsInf(re, 0):
    71  			return complex(math.Copysign(math.Pi/4, re), im)
    72  		default:
    73  			return complex(math.Copysign(0, re), im)
    74  		}
    75  	case math.IsInf(re, 0):
    76  		return complex(math.Copysign(math.Pi/2, re), math.Copysign(re, im))
    77  	}
    78  	ct := complex(-imag(x), real(x)) // i * x
    79  	xx := x * x
    80  	x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x
    81  	x2 := Sqrt(x1)                       // x2 = sqrt(1 - x*x)
    82  	w := Log(ct + x2)
    83  	return complex(imag(w), -real(w)) // -i * w
    84  }
    85  
    86  // Asinh returns the inverse hyperbolic sine of x.
    87  func Asinh(x complex128) complex128 {
    88  	switch re, im := real(x), imag(x); {
    89  	case im == 0 && math.Abs(re) <= 1:
    90  		return complex(math.Asinh(re), im)
    91  	case re == 0 && math.Abs(im) <= 1:
    92  		return complex(re, math.Asin(im))
    93  	case math.IsInf(re, 0):
    94  		switch {
    95  		case math.IsInf(im, 0):
    96  			return complex(re, math.Copysign(math.Pi/4, im))
    97  		case math.IsNaN(im):
    98  			return x
    99  		default:
   100  			return complex(re, math.Copysign(0.0, im))
   101  		}
   102  	case math.IsNaN(re):
   103  		switch {
   104  		case im == 0:
   105  			return x
   106  		case math.IsInf(im, 0):
   107  			return complex(im, re)
   108  		default:
   109  			return NaN()
   110  		}
   111  	case math.IsInf(im, 0):
   112  		return complex(math.Copysign(im, re), math.Copysign(math.Pi/2, im))
   113  	}
   114  	xx := x * x
   115  	x1 := complex(1+real(xx), imag(xx)) // 1 + x*x
   116  	return Log(x + Sqrt(x1))            // log(x + sqrt(1 + x*x))
   117  }
   118  
   119  // Complex circular arc cosine
   120  //
   121  // DESCRIPTION:
   122  //
   123  // w = arccos z  =  PI/2 - arcsin z.
   124  //
   125  // ACCURACY:
   126  //
   127  //                      Relative error:
   128  // arithmetic   domain     # trials      peak         rms
   129  //    DEC       -10,+10      5200      1.6e-15      2.8e-16
   130  //    IEEE      -10,+10     30000      1.8e-14      2.2e-15
   131  
   132  // Acos returns the inverse cosine of x.
   133  func Acos(x complex128) complex128 {
   134  	w := Asin(x)
   135  	return complex(math.Pi/2-real(w), -imag(w))
   136  }
   137  
   138  // Acosh returns the inverse hyperbolic cosine of x.
   139  func Acosh(x complex128) complex128 {
   140  	if x == 0 {
   141  		return complex(0, math.Copysign(math.Pi/2, imag(x)))
   142  	}
   143  	w := Acos(x)
   144  	if imag(w) <= 0 {
   145  		return complex(-imag(w), real(w)) // i * w
   146  	}
   147  	return complex(imag(w), -real(w)) // -i * w
   148  }
   149  
   150  // Complex circular arc tangent
   151  //
   152  // DESCRIPTION:
   153  //
   154  // If
   155  //     z = x + iy,
   156  //
   157  // then
   158  //          1       (    2x     )
   159  // Re w  =  - arctan(-----------)  +  k PI
   160  //          2       (     2    2)
   161  //                  (1 - x  - y )
   162  //
   163  //               ( 2         2)
   164  //          1    (x  +  (y+1) )
   165  // Im w  =  - log(------------)
   166  //          4    ( 2         2)
   167  //               (x  +  (y-1) )
   168  //
   169  // Where k is an arbitrary integer.
   170  //
   171  // catan(z) = -i catanh(iz).
   172  //
   173  // ACCURACY:
   174  //
   175  //                      Relative error:
   176  // arithmetic   domain     # trials      peak         rms
   177  //    DEC       -10,+10      5900       1.3e-16     7.8e-18
   178  //    IEEE      -10,+10     30000       2.3e-15     8.5e-17
   179  // The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
   180  // had peak relative error 1.5e-16, rms relative error
   181  // 2.9e-17.  See also clog().
   182  
   183  // Atan returns the inverse tangent of x.
   184  func Atan(x complex128) complex128 {
   185  	switch re, im := real(x), imag(x); {
   186  	case im == 0:
   187  		return complex(math.Atan(re), im)
   188  	case re == 0 && math.Abs(im) <= 1:
   189  		return complex(re, math.Atanh(im))
   190  	case math.IsInf(im, 0) || math.IsInf(re, 0):
   191  		if math.IsNaN(re) {
   192  			return complex(math.NaN(), math.Copysign(0, im))
   193  		}
   194  		return complex(math.Copysign(math.Pi/2, re), math.Copysign(0, im))
   195  	case math.IsNaN(re) || math.IsNaN(im):
   196  		return NaN()
   197  	}
   198  	x2 := real(x) * real(x)
   199  	a := 1 - x2 - imag(x)*imag(x)
   200  	if a == 0 {
   201  		return NaN()
   202  	}
   203  	t := 0.5 * math.Atan2(2*real(x), a)
   204  	w := reducePi(t)
   205  
   206  	t = imag(x) - 1
   207  	b := x2 + t*t
   208  	if b == 0 {
   209  		return NaN()
   210  	}
   211  	t = imag(x) + 1
   212  	c := (x2 + t*t) / b
   213  	return complex(w, 0.25*math.Log(c))
   214  }
   215  
   216  // Atanh returns the inverse hyperbolic tangent of x.
   217  func Atanh(x complex128) complex128 {
   218  	z := complex(-imag(x), real(x)) // z = i * x
   219  	z = Atan(z)
   220  	return complex(imag(z), -real(z)) // z = -i * z
   221  }
   222
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