1/*-
2 * Copyright (c) 2011 David Schultz
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 *    notice unmodified, this list of conditions, and the following
10 *    disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 *    notice, this list of conditions and the following disclaimer in the
13 *    documentation and/or other materials provided with the distribution.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
25 */
26
27/*
28 * Hyperbolic tangent of a complex argument z = x + i y.
29 *
30 * The algorithm is from:
31 *
32 *   W. Kahan.  Branch Cuts for Complex Elementary Functions or Much
33 *   Ado About Nothing's Sign Bit.  In The State of the Art in
34 *   Numerical Analysis, pp. 165 ff.  Iserles and Powell, eds., 1987.
35 *
36 * Method:
37 *
38 *   Let t    = tan(x)
39 *       beta = 1/cos^2(y)
40 *       s    = sinh(x)
41 *       rho  = cosh(x)
42 *
43 *   We have:
44 *
45 *   tanh(z) = sinh(z) / cosh(z)
46 *
47 *             sinh(x) cos(y) + i cosh(x) sin(y)
48 *           = ---------------------------------
49 *             cosh(x) cos(y) + i sinh(x) sin(y)
50 *
51 *             cosh(x) sinh(x) / cos^2(y) + i tan(y)
52 *           = -------------------------------------
53 *                    1 + sinh^2(x) / cos^2(y)
54 *
55 *             beta rho s + i t
56 *           = ----------------
57 *               1 + beta s^2
58 *
59 * Modifications:
60 *
61 *   I omitted the original algorithm's handling of overflow in tan(x) after
62 *   verifying with nearpi.c that this can't happen in IEEE single or double
63 *   precision.  I also handle large x differently.
64 */
65
66#include <sys/cdefs.h>
67__FBSDID("$FreeBSD$");
68
69#include <complex.h>
70#include <math.h>
71
72#include "math_private.h"
73
74double complex
75ctanh(double complex z)
76{
77	double x, y;
78	double t, beta, s, rho, denom;
79	uint32_t hx, ix, lx;
80
81	x = creal(z);
82	y = cimag(z);
83
84	EXTRACT_WORDS(hx, lx, x);
85	ix = hx & 0x7fffffff;
86
87	/*
88	 * ctanh(NaN + i 0) = NaN + i 0
89	 *
90	 * ctanh(NaN + i y) = NaN + i NaN		for y != 0
91	 *
92	 * The imaginary part has the sign of x*sin(2*y), but there's no
93	 * special effort to get this right.
94	 *
95	 * ctanh(+-Inf +- i Inf) = +-1 +- 0
96	 *
97	 * ctanh(+-Inf + i y) = +-1 + 0 sin(2y)		for y finite
98	 *
99	 * The imaginary part of the sign is unspecified.  This special
100	 * case is only needed to avoid a spurious invalid exception when
101	 * y is infinite.
102	 */
103	if (ix >= 0x7ff00000) {
104		if ((ix & 0xfffff) | lx)	/* x is NaN */
105			return (cpack(x, (y == 0 ? y : x * y)));
106		SET_HIGH_WORD(x, hx - 0x40000000);	/* x = copysign(1, x) */
107		return (cpack(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))));
108	}
109
110	/*
111	 * ctanh(x + i NAN) = NaN + i NaN
112	 * ctanh(x +- i Inf) = NaN + i NaN
113	 */
114	if (!isfinite(y))
115		return (cpack(y - y, y - y));
116
117	/*
118	 * ctanh(+-huge + i +-y) ~= +-1 +- i 2sin(2y)/exp(2x), using the
119	 * approximation sinh^2(huge) ~= exp(2*huge) / 4.
120	 * We use a modified formula to avoid spurious overflow.
121	 */
122	if (ix >= 0x40360000) {	/* x >= 22 */
123		double exp_mx = exp(-fabs(x));
124		return (cpack(copysign(1, x),
125		    4 * sin(y) * cos(y) * exp_mx * exp_mx));
126	}
127
128	/* Kahan's algorithm */
129	t = tan(y);
130	beta = 1.0 + t * t;	/* = 1 / cos^2(y) */
131	s = sinh(x);
132	rho = sqrt(1 + s * s);	/* = cosh(x) */
133	denom = 1 + beta * s * s;
134	return (cpack((beta * rho * s) / denom, t / denom));
135}
136
137double complex
138ctan(double complex z)
139{
140
141	/* ctan(z) = -I * ctanh(I * z) */
142	z = ctanh(cpack(-cimag(z), creal(z)));
143	return (cpack(cimag(z), -creal(z)));
144}
145