1/* origin: OpenBSD /usr/src/lib/libm/src/s_catanl.c */
  2/*
  3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
  4 *
  5 * Permission to use, copy, modify, and distribute this software for any
  6 * purpose with or without fee is hereby granted, provided that the above
  7 * copyright notice and this permission notice appear in all copies.
  8 *
  9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
 10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
 11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
 12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
 13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
 14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
 15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
 16 */
 17/*
 18 *      Complex circular arc tangent
 19 *
 20 *
 21 * SYNOPSIS:
 22 *
 23 * long double complex catanl();
 24 * long double complex z, w;
 25 *
 26 * w = catanl( z );
 27 *
 28 *
 29 * DESCRIPTION:
 30 *
 31 * If
 32 *     z = x + iy,
 33 *
 34 * then
 35 *          1       (    2x     )
 36 * Re w  =  - arctan(-----------)  +  k PI
 37 *          2       (     2    2)
 38 *                  (1 - x  - y )
 39 *
 40 *               ( 2         2)
 41 *          1    (x  +  (y+1) )
 42 * Im w  =  - log(------------)
 43 *          4    ( 2         2)
 44 *               (x  +  (y-1) )
 45 *
 46 * Where k is an arbitrary integer.
 47 *
 48 *
 49 * ACCURACY:
 50 *
 51 *                      Relative error:
 52 * arithmetic   domain     # trials      peak         rms
 53 *    DEC       -10,+10      5900       1.3e-16     7.8e-18
 54 *    IEEE      -10,+10     30000       2.3e-15     8.5e-17
 55 * The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
 56 * had peak relative error 1.5e-16, rms relative error
 57 * 2.9e-17.  See also clog().
 58 */
 59
 60#include <complex.h>
 61#include <float.h>
 62#include "complex_impl.h"
 63
 64#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
 65long double complex catanl(long double complex z)
 66{
 67	return catan(z);
 68}
 69#else
 70static const long double PIL = 3.141592653589793238462643383279502884197169L;
 71static const long double DP1 = 3.14159265358979323829596852490908531763125L;
 72static const long double DP2 = 1.6667485837041756656403424829301998703007e-19L;
 73static const long double DP3 = 1.8830410776607851167459095484560349402753e-39L;
 74
 75static long double redupil(long double x)
 76{
 77	long double t;
 78	long i;
 79
 80	t = x / PIL;
 81	if (t >= 0.0L)
 82		t += 0.5L;
 83	else
 84		t -= 0.5L;
 85
 86	i = t;  /* the multiple */
 87	t = i;
 88	t = ((x - t * DP1) - t * DP2) - t * DP3;
 89	return t;
 90}
 91
 92long double complex catanl(long double complex z)
 93{
 94	long double complex w;
 95	long double a, t, x, x2, y;
 96
 97	x = creall(z);
 98	y = cimagl(z);
 99
100	x2 = x * x;
101	a = 1.0L - x2 - (y * y);
102
103	t = atan2l(2.0L * x, a) * 0.5L;
104	w = redupil(t);
105
106	t = y - 1.0L;
107	a = x2 + (t * t);
108
109	t = y + 1.0L;
110	a = (x2 + (t * t)) / a;
111	w = CMPLXF(w, 0.25L * logl(a));
112	return w;
113}
114#endif