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 "complex_impl.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}