1/* origin: OpenBSD /usr/src/lib/libm/src/s_catan.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 * double complex catan();
 24 * double complex z, w;
 25 *
 26 * w = catan (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 * catan(z) = -i catanh(iz).
 49 *
 50 * ACCURACY:
 51 *
 52 *                      Relative error:
 53 * arithmetic   domain     # trials      peak         rms
 54 *    DEC       -10,+10      5900       1.3e-16     7.8e-18
 55 *    IEEE      -10,+10     30000       2.3e-15     8.5e-17
 56 * The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
 57 * had peak relative error 1.5e-16, rms relative error
 58 * 2.9e-17.  See also clog().
 59 */
 60
 61#include "complex_impl.h"
 62
 63#define MAXNUM 1.0e308
 64
 65static const double DP1 = 3.14159265160560607910E0;
 66static const double DP2 = 1.98418714791870343106E-9;
 67static const double DP3 = 1.14423774522196636802E-17;
 68
 69static double _redupi(double x)
 70{
 71	double t;
 72	long i;
 73
 74	t = x/M_PI;
 75	if (t >= 0.0)
 76		t += 0.5;
 77	else
 78		t -= 0.5;
 79
 80	i = t;  /* the multiple */
 81	t = i;
 82	t = ((x - t * DP1) - t * DP2) - t * DP3;
 83	return t;
 84}
 85
 86double complex catan(double complex z)
 87{
 88	double complex w;
 89	double a, t, x, x2, y;
 90
 91	x = creal(z);
 92	y = cimag(z);
 93
 94	x2 = x * x;
 95	a = 1.0 - x2 - (y * y);
 96
 97	t = 0.5 * atan2(2.0 * x, a);
 98	w = _redupi(t);
 99
100	t = y - 1.0;
101	a = x2 + (t * t);
102
103	t = y + 1.0;
104	a = (x2 + t * t)/a;
105	w = CMPLX(w, 0.25 * log(a));
106	return w;
107}