Re: bin/170206: complex arcsinh, log, etc.

The following reply was made to PR bin/170206; it has been noted by GNATS.

From: Stephen Montgomery-Smith <stephen@missouri.edu>
To: Bruce Evans <brde@optusnet.com.au>
Cc: Stephen Montgomery-Smith <stephen@freebsd.org>,
        FreeBSD-gnats-submit@freebsd.org, freebsd-bugs@freebsd.org
Subject: Re: bin/170206: complex arcsinh, log, etc.
Date: Fri, 27 Jul 2012 09:45:02 -0500

 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.
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