complex atan is not the best it could be

[Stephen Montgomery-Smith <stephen@…>]

For complex atan, when x and y are non-negative, the following formula is rather foolproof:

atanh(z) = 0.25 * log1p(4*x / pow(|z-1|,2)) + 0.5 * I * atan2(2y, (1-x*x-y*y))

where z = x+I*y. Just handle the few special cases when you get overflow or underflow.

I have implemented a C version at ​http://people.freebsd.org/~stephen/catrig.c.

Also, email Tom Fairgrieve <tff@…>. He has an unpublished work by Hull, himself, and Tang, very similar to the work they did on the complex arcsine.

[Stephen Montgomery-Smith <stephen@…>]

Use this for the imaginary part: 0.5 * atan2(2y, (1-x)*(1+x)-y*y) Replacing 1-x*x by (1-x)(1+x) improves the error quite a bit when x is close to 1.

[Stephen Montgomery-Smith <stephen@…>]

The patch I just uploaded is a diff to today's svn. It seems to work rather well.

[John Maddock]

(In [81624]) Lots of small patches. Update and regenerate docs. Fixes #7183. Fixes #7290. Fixes #7291. Fixes #7649. Refs #7492.