Source file src/pkg/math/cmplx/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	//   [email protected]
    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	}