github.com/MangoDowner/go-gm@v0.0.0-20180818020936-8baa2bd4408c/src/math/cmplx/asin.go (about)

     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  	if imag(x) == 0 {
    53  		if math.Abs(real(x)) > 1 {
    54  			return complex(math.Pi/2, 0) // DOMAIN error
    55  		}
    56  		return complex(math.Asin(real(x)), 0)
    57  	}
    58  	ct := complex(-imag(x), real(x)) // i * x
    59  	xx := x * x
    60  	x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x
    61  	x2 := Sqrt(x1)                       // x2 = sqrt(1 - x*x)
    62  	w := Log(ct + x2)
    63  	return complex(imag(w), -real(w)) // -i * w
    64  }
    65  
    66  // Asinh returns the inverse hyperbolic sine of x.
    67  func Asinh(x complex128) complex128 {
    68  	// TODO check range
    69  	if imag(x) == 0 {
    70  		if math.Abs(real(x)) > 1 {
    71  			return complex(math.Pi/2, 0) // DOMAIN error
    72  		}
    73  		return complex(math.Asinh(real(x)), 0)
    74  	}
    75  	xx := x * x
    76  	x1 := complex(1+real(xx), imag(xx)) // 1 + x*x
    77  	return Log(x + Sqrt(x1))            // log(x + sqrt(1 + x*x))
    78  }
    79  
    80  // Complex circular arc cosine
    81  //
    82  // DESCRIPTION:
    83  //
    84  // w = arccos z  =  PI/2 - arcsin z.
    85  //
    86  // ACCURACY:
    87  //
    88  //                      Relative error:
    89  // arithmetic   domain     # trials      peak         rms
    90  //    DEC       -10,+10      5200      1.6e-15      2.8e-16
    91  //    IEEE      -10,+10     30000      1.8e-14      2.2e-15
    92  
    93  // Acos returns the inverse cosine of x.
    94  func Acos(x complex128) complex128 {
    95  	w := Asin(x)
    96  	return complex(math.Pi/2-real(w), -imag(w))
    97  }
    98  
    99  // Acosh returns the inverse hyperbolic cosine of x.
   100  func Acosh(x complex128) complex128 {
   101  	w := Acos(x)
   102  	if imag(w) <= 0 {
   103  		return complex(-imag(w), real(w)) // i * w
   104  	}
   105  	return complex(imag(w), -real(w)) // -i * w
   106  }
   107  
   108  // Complex circular arc tangent
   109  //
   110  // DESCRIPTION:
   111  //
   112  // If
   113  //     z = x + iy,
   114  //
   115  // then
   116  //          1       (    2x     )
   117  // Re w  =  - arctan(-----------)  +  k PI
   118  //          2       (     2    2)
   119  //                  (1 - x  - y )
   120  //
   121  //               ( 2         2)
   122  //          1    (x  +  (y+1) )
   123  // Im w  =  - log(------------)
   124  //          4    ( 2         2)
   125  //               (x  +  (y-1) )
   126  //
   127  // Where k is an arbitrary integer.
   128  //
   129  // catan(z) = -i catanh(iz).
   130  //
   131  // ACCURACY:
   132  //
   133  //                      Relative error:
   134  // arithmetic   domain     # trials      peak         rms
   135  //    DEC       -10,+10      5900       1.3e-16     7.8e-18
   136  //    IEEE      -10,+10     30000       2.3e-15     8.5e-17
   137  // The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
   138  // had peak relative error 1.5e-16, rms relative error
   139  // 2.9e-17.  See also clog().
   140  
   141  // Atan returns the inverse tangent of x.
   142  func Atan(x complex128) complex128 {
   143  	if real(x) == 0 && imag(x) > 1 {
   144  		return NaN()
   145  	}
   146  
   147  	x2 := real(x) * real(x)
   148  	a := 1 - x2 - imag(x)*imag(x)
   149  	if a == 0 {
   150  		return NaN()
   151  	}
   152  	t := 0.5 * math.Atan2(2*real(x), a)
   153  	w := reducePi(t)
   154  
   155  	t = imag(x) - 1
   156  	b := x2 + t*t
   157  	if b == 0 {
   158  		return NaN()
   159  	}
   160  	t = imag(x) + 1
   161  	c := (x2 + t*t) / b
   162  	return complex(w, 0.25*math.Log(c))
   163  }
   164  
   165  // Atanh returns the inverse hyperbolic tangent of x.
   166  func Atanh(x complex128) complex128 {
   167  	z := complex(-imag(x), real(x)) // z = i * x
   168  	z = Atan(z)
   169  	return complex(imag(z), -real(z)) // z = -i * z
   170  }