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
5package cmplx
6
7import "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 sine
32//
33// DESCRIPTION:
34//
35// If
36//     z = x + iy,
37//
38// then
39//
40//     w = sin x  cosh y  +  i cos x sinh y.
41//
42// csin(z) = -i csinh(iz).
43//
44// ACCURACY:
45//
46//                      Relative error:
47// arithmetic   domain     # trials      peak         rms
48//    DEC       -10,+10      8400       5.3e-17     1.3e-17
49//    IEEE      -10,+10     30000       3.8e-16     1.0e-16
50// Also tested by csin(casin(z)) = z.
51
52// Sin returns the sine of x.
53func Sin(x complex128) complex128 {
54	switch re, im := real(x), imag(x); {
55	case im == 0 && (math.IsInf(re, 0) || math.IsNaN(re)):
56		return complex(math.NaN(), im)
57	case math.IsInf(im, 0):
58		switch {
59		case re == 0:
60			return x
61		case math.IsInf(re, 0) || math.IsNaN(re):
62			return complex(math.NaN(), im)
63		}
64	case re == 0 && math.IsNaN(im):
65		return x
66	}
67	s, c := math.Sincos(real(x))
68	sh, ch := sinhcosh(imag(x))
69	return complex(s*ch, c*sh)
70}
71
72// Complex hyperbolic sine
73//
74// DESCRIPTION:
75//
76// csinh z = (cexp(z) - cexp(-z))/2
77//         = sinh x * cos y  +  i cosh x * sin y .
78//
79// ACCURACY:
80//
81//                      Relative error:
82// arithmetic   domain     # trials      peak         rms
83//    IEEE      -10,+10     30000       3.1e-16     8.2e-17
84
85// Sinh returns the hyperbolic sine of x.
86func Sinh(x complex128) complex128 {
87	switch re, im := real(x), imag(x); {
88	case re == 0 && (math.IsInf(im, 0) || math.IsNaN(im)):
89		return complex(re, math.NaN())
90	case math.IsInf(re, 0):
91		switch {
92		case im == 0:
93			return complex(re, im)
94		case math.IsInf(im, 0) || math.IsNaN(im):
95			return complex(re, math.NaN())
96		}
97	case im == 0 && math.IsNaN(re):
98		return complex(math.NaN(), im)
99	}
100	s, c := math.Sincos(imag(x))
101	sh, ch := sinhcosh(real(x))
102	return complex(c*sh, s*ch)
103}
104
105// Complex circular cosine
106//
107// DESCRIPTION:
108//
109// If
110//     z = x + iy,
111//
112// then
113//
114//     w = cos x  cosh y  -  i sin x sinh y.
115//
116// ACCURACY:
117//
118//                      Relative error:
119// arithmetic   domain     # trials      peak         rms
120//    DEC       -10,+10      8400       4.5e-17     1.3e-17
121//    IEEE      -10,+10     30000       3.8e-16     1.0e-16
122
123// Cos returns the cosine of x.
124func Cos(x complex128) complex128 {
125	switch re, im := real(x), imag(x); {
126	case im == 0 && (math.IsInf(re, 0) || math.IsNaN(re)):
127		return complex(math.NaN(), -im*math.Copysign(0, re))
128	case math.IsInf(im, 0):
129		switch {
130		case re == 0:
131			return complex(math.Inf(1), -re*math.Copysign(0, im))
132		case math.IsInf(re, 0) || math.IsNaN(re):
133			return complex(math.Inf(1), math.NaN())
134		}
135	case re == 0 && math.IsNaN(im):
136		return complex(math.NaN(), 0)
137	}
138	s, c := math.Sincos(real(x))
139	sh, ch := sinhcosh(imag(x))
140	return complex(c*ch, -s*sh)
141}
142
143// Complex hyperbolic cosine
144//
145// DESCRIPTION:
146//
147// ccosh(z) = cosh x  cos y + i sinh x sin y .
148//
149// ACCURACY:
150//
151//                      Relative error:
152// arithmetic   domain     # trials      peak         rms
153//    IEEE      -10,+10     30000       2.9e-16     8.1e-17
154
155// Cosh returns the hyperbolic cosine of x.
156func Cosh(x complex128) complex128 {
157	switch re, im := real(x), imag(x); {
158	case re == 0 && (math.IsInf(im, 0) || math.IsNaN(im)):
159		return complex(math.NaN(), re*math.Copysign(0, im))
160	case math.IsInf(re, 0):
161		switch {
162		case im == 0:
163			return complex(math.Inf(1), im*math.Copysign(0, re))
164		case math.IsInf(im, 0) || math.IsNaN(im):
165			return complex(math.Inf(1), math.NaN())
166		}
167	case im == 0 && math.IsNaN(re):
168		return complex(math.NaN(), im)
169	}
170	s, c := math.Sincos(imag(x))
171	sh, ch := sinhcosh(real(x))
172	return complex(c*ch, s*sh)
173}
174
175// calculate sinh and cosh
176func sinhcosh(x float64) (sh, ch float64) {
177	if math.Abs(x) <= 0.5 {
178		return math.Sinh(x), math.Cosh(x)
179	}
180	e := math.Exp(x)
181	ei := 0.5 / e
182	e *= 0.5
183	return e - ei, e + ei
184}
185