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1 /* origin: FreeBSD /usr/src/lib/msun/src/s_ctanh.c */
3 * Copyright (c) 2011 David Schultz
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
9 * 1. Redistributions of source code must retain the above copyright
10 * notice unmodified, this list of conditions, and the following
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.
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.
28 * Hyperbolic tangent of a complex argument z = x + i y.
30 * The algorithm is from:
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.
45 * tanh(z) = sinh(z) / cosh(z)
47 * sinh(x) cos(y) + i cosh(x) sin(y)
48 * = ---------------------------------
49 * cosh(x) cos(y) + i sinh(x) sin(y)
51 * cosh(x) sinh(x) / cos^2(y) + i tan(y)
52 * = -------------------------------------
53 * 1 + sinh^2(x) / cos^2(y)
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.
68 double complex ctanh(double complex z
)
71 double t
, beta
, s
, rho
, denom
;
77 EXTRACT_WORDS(hx
, lx
, x
);
81 * ctanh(NaN + i 0) = NaN + i 0
83 * ctanh(NaN + i y) = NaN + i NaN for y != 0
85 * The imaginary part has the sign of x*sin(2*y), but there's no
86 * special effort to get this right.
88 * ctanh(+-Inf +- i Inf) = +-1 +- 0
90 * ctanh(+-Inf + i y) = +-1 + 0 sin(2y) for y finite
92 * The imaginary part of the sign is unspecified. This special
93 * case is only needed to avoid a spurious invalid exception when
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
)));
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
110 return CMPLX(x
? y
- y
: x
, y
- y
);
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.
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
);
122 /* Kahan's algorithm */
124 beta
= 1.0 + t
* t
; /* = 1 / cos^2(y) */
126 rho
= sqrt(1 + s
* s
); /* = cosh(x) */
127 denom
= 1 + beta
* s
* s
;
128 return CMPLX((beta
* rho
* s
) / denom
, t
/ denom
);