On Wednesday, July 29, 2015 at 5:31:12 PM UTC-4, Stefan Karpinski wrote: > > The most compelling part of the proposal to me was the claim of > associativity, which I suppose comes along with the variable precision > since you can actually drop trailing bits that you can't get right. >
I bought a copy of the book, because I'm a sucker for this kind of thing. There's a lot of fascinating material in the book, and I would generally recommend it, with the caveat that it seems like some of it needs to be taken with a grain of salt. Remember that it's a manuscript that hasn't gone through the kind of peer review that journal articles do. Associativity sounded pretty exciting to me, too, but you have to do special work to get it. If a, b, and c are unums or ubounds, it is *not* the case that you will always have (a+b)+c=a+(b+c), if you write the calculation that way. Like other kinds of interval arithmetic, ubounds obey sub-associativity, which says that the two sides of that equation are "not nowhere equal", and that their intersection contains the exact answer. The way you get associativity is by using a fused sum operation that internally accumulates sums with enough precision to restore associativity of the rounded end results. Here's a page from the book: https://books.google.com/books?id=fZsXBgAAQBAJ&pg=PA164&lpg=PA164&dq=subassociative+interval+arithmetic&source=bl&ots=MH0veKCw9Y&sig=sq9atiYn46w2rvuT26E3fLbUnOg&hl=en&sa=X&ved=0CDYQ6AEwA2oVChMIi7Dyt6eDxwIVA24-Ch3gjg-7#v=onepage&q=subassociative%20interval%20arithmetic&f=false Gustafson's whole proposal involves standardizing several layers of computation, including what happens in a higher precision scratchpad that is imagined to be in hardware and fenced off from the user. IEEE floating point arithmetic also works with a higher precision scratchpad, but exactly what happens there is a little bit under constrained, and has varied between different processors. Standardizing the scratchpad, and which fused operations will keep their operands there, seems to be an important part of the proposal. I'm pretty interested in more discussion of the book, but this mailing list probably isn't the right place for a wide ranging discussion to happen. Does anyone have any advice about other appropriate forums?
