Date: Sat, 28 Jul 2012 18:12:07 -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: <501471C7.8000102@missouri.edu> In-Reply-To: <50141018.3040203@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> <50141018.3040203@missouri.edu>
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On 07/28/2012 11:15 AM, Stephen Montgomery-Smith wrote: > 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.) And I should add that I just realized that ULP isn't quite the same as relative error, so an extra factor of up to 2 could make its way into the calculations.
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