s_ctanh.c revision 330897 
1/*-
2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
3 *
4 * Copyright (c) 2011 David Schultz
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 *    notice unmodified, this list of conditions, and the following
12 *    disclaimer.
13 * 2. Redistributions in binary form must reproduce the above copyright
14 *    notice, this list of conditions and the following disclaimer in the
15 *    documentation and/or other materials provided with the distribution.
16 *
17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 */
28
29/*
30 * Hyperbolic tangent of a complex argument z = x + I y.
31 *
32 * The algorithm is from:
33 *
34 *   W. Kahan.  Branch Cuts for Complex Elementary Functions or Much
35 *   Ado About Nothing's Sign Bit.  In The State of the Art in
36 *   Numerical Analysis, pp. 165 ff.  Iserles and Powell, eds., 1987.
37 *
38 * Method:
39 *
40 *   Let t    = tan(x)
41 *       beta = 1/cos^2(y)
42 *       s    = sinh(x)
43 *       rho  = cosh(x)
44 *
45 *   We have:
46 *
47 *   tanh(z) = sinh(z) / cosh(z)
48 *
49 *             sinh(x) cos(y) + I cosh(x) sin(y)
50 *           = ---------------------------------
51 *             cosh(x) cos(y) + I sinh(x) sin(y)
52 *
53 *             cosh(x) sinh(x) / cos^2(y) + I tan(y)
54 *           = -------------------------------------
55 *                    1 + sinh^2(x) / cos^2(y)
56 *
57 *             beta rho s + I t
58 *           = ----------------
59 *               1 + beta s^2
60 *
61 * Modifications:
62 *
63 *   I omitted the original algorithm's handling of overflow in tan(x) after
64 *   verifying with nearpi.c that this can't happen in IEEE single or double
65 *   precision.  I also handle large x differently.
66 */
67
68#include <sys/cdefs.h>
69__FBSDID("$FreeBSD: stable/11/lib/msun/src/s_ctanh.c 330897 2018-03-14 03:19:51Z eadler $");
70
71#include <complex.h>
72#include <math.h>
73
74#include "math_private.h"
75
76double complex
77ctanh(double complex z)
78{
79	double x, y;
80	double t, beta, s, rho, denom;
81	uint32_t hx, ix, lx;
82
83	x = creal(z);
84	y = cimag(z);
85
86	EXTRACT_WORDS(hx, lx, x);
87	ix = hx & 0x7fffffff;
88
89	/*
90	 * ctanh(NaN +- I 0) = d(NaN) +- I 0
91	 *
92	 * ctanh(NaN + I y) = d(NaN,y) + I d(NaN,y)	for y != 0
93	 *
94	 * The imaginary part has the sign of x*sin(2*y), but there's no
95	 * special effort to get this right.
96	 *
97	 * ctanh(+-Inf +- I Inf) = +-1 +- I 0
98	 *
99	 * ctanh(+-Inf + I y) = +-1 + I 0 sin(2y)	for y finite
100	 *
101	 * The imaginary part of the sign is unspecified.  This special
102	 * case is only needed to avoid a spurious invalid exception when
103	 * y is infinite.
104	 */
105	if (ix >= 0x7ff00000) {
106		if ((ix & 0xfffff) | lx)	/* x is NaN */
107			return (CMPLX((x + 0) * (y + 0),
108			    y == 0 ? y : (x + 0) * (y + 0)));
109		SET_HIGH_WORD(x, hx - 0x40000000);	/* x = copysign(1, x) */
110		return (CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))));
111	}
112
113	/*
114	 * ctanh(x + I NaN) = d(NaN) + I d(NaN)
115	 * ctanh(x +- I Inf) = dNaN + I dNaN
116	 */
117	if (!isfinite(y))
118		return (CMPLX(y - y, y - y));
119
120	/*
121	 * ctanh(+-huge +- I y) ~= +-1 +- I 2sin(2y)/exp(2x), using the
122	 * approximation sinh^2(huge) ~= exp(2*huge) / 4.
123	 * We use a modified formula to avoid spurious overflow.
124	 */
125	if (ix >= 0x40360000) {	/* |x| >= 22 */
126		double exp_mx = exp(-fabs(x));
127		return (CMPLX(copysign(1, x),
128		    4 * sin(y) * cos(y) * exp_mx * exp_mx));
129	}
130
131	/* Kahan's algorithm */
132	t = tan(y);
133	beta = 1.0 + t * t;	/* = 1 / cos^2(y) */
134	s = sinh(x);
135	rho = sqrt(1 + s * s);	/* = cosh(x) */
136	denom = 1 + beta * s * s;
137	return (CMPLX((beta * rho * s) / denom, t / denom));
138}
139
140double complex
141ctan(double complex z)
142{
143
144	/* ctan(z) = -I * ctanh(I * z) = I * conj(ctanh(I * conj(z))) */
145	z = ctanh(CMPLX(cimag(z), creal(z)));
146	return (CMPLX(cimag(z), creal(z)));
147}
148