src/complex/catan.c

   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
  65 static const double DP1 = 3.14159265160560607910E0;
  66 static const double DP2 = 1.98418714791870343106E-9;
  67 static const double DP3 = 1.14423774522196636802E-17;
  68
  69 static 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
  86 double 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 }