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Message-ID: <cc6f1778-1281-e9e4-31f8-3abb39e6b7e8@esi.com.au>
Date: Fri, 31 Jul 2026 17:39:45 +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
As far as I can tell looking at old implementations of acosh(x), that this
approximation was never visited for x is negative because an x is
negative case was handled earlier in the routines.
I'm not even clear why the former form is preferred for
medium-magnitude positive x; it doesn't seem to be avoiding any
overflow or cancellation problems.
At the risk of oversimplifying it, I think because the error in the term
1/(x + sqrt(x * x - 1))
relative to 2 * x is smaller (in an overall sense) than the error in the
term
sqrt(x * x - 1)
relative to x.
That said, one newer implementation seeks better accuracy and instead
breaks the domain [2..128] into multiple segments and then approximates
each with a minimax polynomial. Scary stuff. AMD's AOCL.
Another by Sibidanov does something similar but it is really scary stuff
and I am eagerly awaiting the documentation because I cannot understand
much of it. Maybe with documentation I won't understand much more. GLIBC.
Thanks - Damian
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