bin/170206: complex arcsinh, log, etc.

[Stephen Montgomery-Smith]

On 07/27/2012 09:26 AM, Bruce Evans wrote:

> %     hm1 = -1;
> %     for (i=0;i<12;i++) hm1 += val[i];
> %     return (cpack(0.5 * log1p(hm1), atan2(y, x)));
>
> It is the trailing terms that I think don't work right here.  You sort
> them and add from high to low, but normally it is necessary to add
> from low to high (consider terms [1, DBL_EPSILON/2, DBL_EPSILON/4]).
> Adding from high to low cancels with the -1 term, but then only
> particular values work right.  Also, I don't see how adding the low
> terms without extra precision preserves enough precision.

I understand what you are saying.  But in this case adding in order of 
smallest to largest (adding -1 last) won't work.  If all the signs in 
the same direction, it would work.  But -1 has the wrong sign.

But I can also tell you that I haven't thought my algorithm through 
every special case.  I can tell you it seems to work in all the examples 
I tried.  But I don't have a mathematical proof.