text: /*-
   * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
   *
   * Copyright (c) 2011 David Schultz
   * All rights reserved.
   *
   * Redistribution and use in source and binary forms, with or without
   * modification, are permitted provided that the following conditions
   * are met:
   * 1. Redistributions of source code must retain the above copyright
   *    notice unmodified, this list of conditions, and the following
   *    disclaimer.
   * 2. Redistributions in binary form must reproduce the above copyright
   *    notice, this list of conditions and the following disclaimer in the
   *    documentation and/or other materials provided with the distribution.
   *
   * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
   * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
   * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
   * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
   * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
   * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
   * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
   * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
   * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
   * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
   */
  
  /*
   * Hyperbolic tangent of a complex argument z = x + I y.
   *
   * The algorithm is from:
   *
   *   W. Kahan.  Branch Cuts for Complex Elementary Functions or Much
   *   Ado About Nothing's Sign Bit.  In The State of the Art in
   *   Numerical Analysis, pp. 165 ff.  Iserles and Powell, eds., 1987.
   *
   * Method:
   *
   *   Let t    = tan(x)
   *       beta = 1/cos^2(y)
   *       s    = sinh(x)
   *       rho  = cosh(x)
   *
   *   We have:
   *
   *   tanh(z) = sinh(z) / cosh(z)
   *
   *             sinh(x) cos(y) + I cosh(x) sin(y)
   *           = ---------------------------------
   *             cosh(x) cos(y) + I sinh(x) sin(y)
   *
   *             cosh(x) sinh(x) / cos^2(y) + I tan(y)
   *           = -------------------------------------
   *                    1 + sinh^2(x) / cos^2(y)
   *
   *             beta rho s + I t
   *           = ----------------
   *               1 + beta s^2
   *
   * Modifications:
   *
   *   I omitted the original algorithm's handling of overflow in tan(x) after
   *   verifying with nearpi.c that this can't happen in IEEE single or double
   *   precision.  I also handle large x differently.
   */
  
  #include <sys/cdefs.h>
  
  #include <complex.h>
  #include <math.h>
  
  #include "math_private.h"
  
  double complex
  ctanh(double complex z)
  {
  	double x, y;
  	double t, beta, s, rho, denom;
  	uint32_t hx, ix, lx;
  
  	x = creal(z);
  	y = cimag(z);
  
  	EXTRACT_WORDS(hx, lx, x);
  	ix = hx & 0x7fffffff;
  
  	/*
  	 * ctanh(NaN +- I 0) = d(NaN) +- I 0
  	 *
  	 * ctanh(NaN + I y) = d(NaN,y) + I d(NaN,y)	for y != 0
  	 *
  	 * The imaginary part has the sign of x*sin(2*y), but there's no
  	 * special effort to get this right.
  	 *
  	 * ctanh(+-Inf +- I Inf) = +-1 +- I 0
  	 *
  	 * ctanh(+-Inf + I y) = +-1 + I 0 sin(2y)	for y finite
  	 *
  	 * The imaginary part of the sign is unspecified.  This special
  	 * case is only needed to avoid a spurious invalid exception when
  	 * y is infinite.
  	 */
  	if (ix >= 0x7ff00000) {
  		if ((ix & 0xfffff) | lx)	/* x is NaN */
  			return (CMPLX(nan_mix(x, y),
  			    y == 0 ? y : nan_mix(x, y)));
  		SET_HIGH_WORD(x, hx - 0x40000000);	/* x = copysign(1, x) */
  		return (CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))));
  	}
  
  	/*
  	 * ctanh(x + I NaN) = d(NaN) + I d(NaN)
  	 * ctanh(x +- I Inf) = dNaN + I dNaN
  	 */
  	if (!isfinite(y))
  		return (CMPLX(y - y, y - y));
  
  	/*
  	 * ctanh(+-huge +- I y) ~= +-1 +- I 2sin(2y)/exp(2x), using the
  	 * approximation sinh^2(huge) ~= exp(2*huge) / 4.
  	 * We use a modified formula to avoid spurious overflow.
  	 */
  	if (ix >= 0x40360000) {	/* |x| >= 22 */
  		double exp_mx = exp(-fabs(x));
  		return (CMPLX(copysign(1, x),
  		    4 * sin(y) * cos(y) * exp_mx * exp_mx));
  	}
  
  	/* Kahan's algorithm */
  	t = tan(y);
  	beta = 1.0 + t * t;	/* = 1 / cos^2(y) */
  	s = sinh(x);
  	rho = sqrt(1 + s * s);	/* = cosh(x) */
  	denom = 1 + beta * s * s;
  	return (CMPLX((beta * rho * s) / denom, t / denom));
  }
  
  double complex
  ctan(double complex z)
  {
  
  	/* ctan(z) = -I * ctanh(I * z) = I * conj(ctanh(I * conj(z))) */
  	z = ctanh(CMPLX(cimag(z), creal(z)));
  	return (CMPLX(cimag(z), creal(z)));
  }
  