On Thursday, July 30, 2015 at 4:22:34 PM UTC-4, Job van der Zwan wrote: > > On Thursday, 30 July 2015 21:54:39 UTC+2, Jason Merrill wrote: > >> <Analysis of examples in the book> >> > > Thanks for correcting me! The open/closed element becomes pretty crucial > later on though, when he claims on page 225 that: > > a general approach for evaluating polynomials with interval arguments >> without any information loss is presented here for the first time. >> > > Two pages later he gives the general scheme for it (see attached picture - > it was too much of a pain to extract that text with proper formatting. This > is ok under fair use right?). > > Do you have any thoughts on that? >
The fused polynomial evaluation seems pretty brilliant to me. He later goes on to suggest having a fused product ratio, which should largely allow eliminating the dependency problem from evaluating rational functions. You can get an awful lot done with rational functions. <https://lh3.googleusercontent.com/-f-sYnCMJFpQ/VbqE8zbN5AI/AAAAAAAAHOk/cNTnxAUAyoU/s1600/polynomial.png>I actually think keeping track of open vs. closed intervals sounds like a pretty good idea. It might also be worth doing for other kinds of interval arithmetic, and I don't see any major reason that that would be impossible. I didn't meant to say that open vs closed intervals doesn't matter--I just meant that it doesn't seem to be the "secret sauce" in any of the challenge problems in Chapter 14. To me, the fused operations are the secret sauce in terms of precision, and the variable length representation *might be* the secret sauce for performance, but I can't really comment on that.
