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Message-ID: <2a413619-3367-591c-d3ee-5417656185a@esi.com.au>
Date: Sat, 1 Aug 2026 08:55:28 +1000 (AEST)
From: Damian McGuckin <damianm@....com.au>
To: musl@...ts.openwall.com
cc: Paul Zimmermann <Paul.Zimmermann@...ia.fr>
Subject: Re: issue in acosh
On Fri, 31 Jul 2026, Rich Felker wrote:
> The problem looks to be that, when x is negative,
>
> 2*x - 1/(x+sqrt(x*x-1))
>
> is a very inaccurate way of computing the desired value
>
> x + sqrt(x*x-1)
>
> I'm not even clear why the former form is preferred for
> medium-magnitude positive x;
At worst, it buys one extra bit of accuracy in the argument to log(). On
average, it buys 3 bits extra. In terms of the result, the error drops
from about 0.92ULP down to about 0.75ULP which I consider acceptable but a
long way from correctly rounded if you need that.
Then again, the computation for x in [1,2) which uses log1p() and is
effectively
log(x + sqrt(x*x-1))
which still has a 32-bit relative error of about 1.09ULP even if you use
an FMA which is not so flash. I do not know how to get it significantly
under 1*ULP although there are at least two approaches which yield a
correctly rounded result (at a significant increase in complexity).
Thanks - Damian
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