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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