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      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: head/lib/msun/src/s_ctanh.c 336362 2018-07-17 07:42:14Z bde $");
     70 
     71 #include <complex.h>
     72 #include <math.h>
     73 
     74 #include "math_private.h"
     75 
     76 double complex
     77 ctanh(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(nan_mix(x, y),
    108 			    y == 0 ? y : nan_mix(x, y)));
    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 
    140 double complex
    141 ctan(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