src/complex/catanl.c

   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
  65 long double complex catanl(long double complex z)
  66 {
  67         return catan(z);
  68 }
  69 #else
  70 static const long double PIL = 3.141592653589793238462643383279502884197169L;
  71 static const long double DP1 = 3.14159265358979323829596852490908531763125L;
  72 static const long double DP2 = 1.6667485837041756656403424829301998703007e-19L;
  73 static const long double DP3 = 1.8830410776607851167459095484560349402753e-39L;
  74
  75 static 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
  92 long 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