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