I discuss this in the book; there have to be strict bounds on how long a computation remains in the *g*-layer (fused) or people would dump their entire calculation in there. I think i got most of the fused operations that make sense, and I pointed out some that do not make sense. It is key that you should have a finite and predictable bound on the memory requirement of the *g*-layer where scratch work is done. It cannot be regarded as unlimited, or limited only by available system memory. For every fused operation, I can predict how many bits will be needed to return a correct answer, which means there is hope for a hardware implementation someday.
On Thursday, July 30, 2015 at 3:44:26 PM UTC-7, Stefan Karpinski wrote: > > Fused polynomials do seem like a good idea (again, can be done for > intervals too), but what is the end game of this approach? Is there some > set of primitives that are sufficient to express all computations you might > want to do in a way that doesn't lose accuracy too rapidly to be useful? It > seems like the reductio ad absurdum is producing a fused version of your > entire program that cleverly produces a correct interval. > > On Thu, Jul 30, 2015 at 5:20 PM, Jason Merrill <[email protected] > <javascript:>> wrote: > >> 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. >> > >
