catan

Complex circular arc tangent.

SYNOPSIS:

void catan();
cmplx z, w;

catan(&z, &w);

DESCRIPTION:

If
     z = x + iy,

then
         1       (     2x      )
Re w = - arctan( ----------- ) + k PI
         2       (      2    2 )
                 ( 1 - x - y )

              ( 2         2 )
         1    ( x + (y+1) )
Im w = - log( ------------ )
         4    ( 2         2 )
              ( x + (y-1) )

Where k is an arbitrary integer.

ACCURACY:

                      Relative error:
arithmetic   domain     # trials      peak         rms
    DEC       -10,+10      5900       1.3e-16     7.8e-18
    IEEE      -10,+10     30000       2.3e-15     8.5e-17

The check catan(ctan(z)) = z, with |x| and |y| < PI/2, had peak relative error 1.5e-16, rms relative error 2.9e-17. See also clog().