# Source file src/math/cmplx/asin.go

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
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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