1/* origin: FreeBSD /usr/src/lib/msun/src/s_ctanh.c */
2/*-
3 * Copyright (c) 2011 David Schultz
4 * All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 *    notice unmodified, this list of conditions, and the following
11 *    disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 *    notice, this list of conditions and the following disclaimer in the
14 *    documentation and/or other materials provided with the distribution.
15 *
16 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
17 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
18 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
19 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
20 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
21 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
22 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
23 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
24 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
25 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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 "libm.h"
67
68double complex ctanh(double complex z)
69{
70	double x, y;
71	double t, beta, s, rho, denom;
72	uint32_t hx, ix, lx;
73
74	x = creal(z);
75	y = cimag(z);
76
77	EXTRACT_WORDS(hx, lx, x);
78	ix = hx & 0x7fffffff;
79
80	/*
81	 * ctanh(NaN + i 0) = NaN + i 0
82	 *
83	 * ctanh(NaN + i y) = NaN + i NaN               for y != 0
84	 *
85	 * The imaginary part has the sign of x*sin(2*y), but there's no
86	 * special effort to get this right.
87	 *
88	 * ctanh(+-Inf +- i Inf) = +-1 +- 0
89	 *
90	 * ctanh(+-Inf + i y) = +-1 + 0 sin(2y)         for y finite
91	 *
92	 * The imaginary part of the sign is unspecified.  This special
93	 * case is only needed to avoid a spurious invalid exception when
94	 * y is infinite.
95	 */
96	if (ix >= 0x7ff00000) {
97		if ((ix & 0xfffff) | lx)        /* x is NaN */
98			return CMPLX(x, (y == 0 ? y : x * y));
99		SET_HIGH_WORD(x, hx - 0x40000000);      /* x = copysign(1, x) */
100		return CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y)));
101	}
102
103	/*
104	 * ctanh(+-0 + i NAN) = +-0 + i NaN
105	 * ctanh(+-0 +- i Inf) = +-0 + i NaN
106	 * ctanh(x + i NAN) = NaN + i NaN
107	 * ctanh(x +- i Inf) = NaN + i NaN
108	 */
109	if (!isfinite(y))
110		return CMPLX(x ? y - y : x, y - y);
111
112	/*
113	 * ctanh(+-huge + i +-y) ~= +-1 +- i 2sin(2y)/exp(2x), using the
114	 * approximation sinh^2(huge) ~= exp(2*huge) / 4.
115	 * We use a modified formula to avoid spurious overflow.
116	 */
117	if (ix >= 0x40360000) { /* x >= 22 */
118		double exp_mx = exp(-fabs(x));
119		return CMPLX(copysign(1, x), 4 * sin(y) * cos(y) * exp_mx * exp_mx);
120	}
121
122	/* Kahan's algorithm */
123	t = tan(y);
124	beta = 1.0 + t * t;     /* = 1 / cos^2(y) */
125	s = sinh(x);
126	rho = sqrt(1 + s * s);  /* = cosh(x) */
127	denom = 1 + beta * s * s;
128	return CMPLX((beta * rho * s) / denom, t / denom);
129}
130