]>
Commit | Line | Data |
---|---|---|
320054e8 DG |
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 | ||
d4db3fa2 | 66 | #include "complex_impl.h" |
320054e8 DG |
67 | |
68 | double 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 | } |