Date: Sat, 28 Jul 2012 11:15:20 -0500 From: Stephen Montgomery-Smith <stephen@missouri.edu> To: Bruce Evans <brde@optusnet.com.au> Cc: freebsd-bugs@freebsd.org, FreeBSD-gnats-submit@freebsd.org, Stephen Montgomery-Smith <stephen@freebsd.org> Subject: Re: bin/170206: complex arcsinh, log, etc. Message-ID: <50141018.3040203@missouri.edu> In-Reply-To: <50140956.1030603@missouri.edu> References: <201207270247.q6R2lkeR021134@wilberforce.math.missouri.edu> <20120727233939.A7820@besplex.bde.org> <5012A96E.9090400@missouri.edu> <20120728142915.K909@besplex.bde.org> <50137C24.1060004@missouri.edu> <20120728171345.T1911@besplex.bde.org> <50140956.1030603@missouri.edu>
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On 07/28/2012 10:46 AM, Stephen Montgomery-Smith wrote:
> OK. This clog really seems to work.
>
> x*x + y*y - 1 is computed with a ULP less than 0.8. The rest of the
> errors seem to be due to the implementation of log1p. The ULP of the
> final answer seems to be never bigger than a little over 2.
>
>
Also, I don't think the problem is due to the implementation of log1p.
If you do an error analysis of log(1+x) where x is about exp(-1)-1, and
x is correct to within 0.8 ULP, I suspect that about 2.5 ULP is the best
you can do for the final answer:
relative_error(log(1+x)) = fabs(1/((1+x) log(1+x))) * relative_error(x)
= 1.58 * relative_error(x)
Given that log1p has itself a ULP of about 1, and relative error in x is
0.8, and considering x=exp(-1)-1, this gives a ULP at around 1.58*0.8+1
= 2.3. And that is what I observed.
(Here "=" means approximately equal to.)
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