github.com/afumu/libc@v0.0.6/musl/src/complex/ctanh.c (about)

     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  
    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  }