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I don't think this can do any of the "standard" constants or what we generally consider to be closed-form expressions, though ! (E.g., no e, pi, exp, log, etc.)
by LolWolf 5mo ago
I don't think this can do any of the "standard" constants or what we generally consider to be closed-form expressions, though ! (E.g., no e, pi, exp, log, etc.)
- SideQuark 5mo agoYes it can, by using the same infinite series that exp and ln use to compute. This one just costs less in money, hardware, energy, and is faster for basically every basic op.
- LolWolf 5mo agoI think the point is that it is _finite_. if you allow infinite expressions then the basic monomial basis or quotients thereof are “even simpler”
- SideQuark 5mo agoIt’s only finite by putting the infinite series into an operation. And the basic monomial basis is not a single binary operation capable of reproducing the set of basic arithmetic ops. If you want trivial and basic, pick Peano postulates. But that’s not what this thread was about.
- LolWolf 5mo agowell, the statement is: is there a single operation, built from elementary operations, such that all _other_ elementary operations have finite representations. this preprint answers that in the affirmative otoh, (x, y) -> 1/(x-y) does not answer this question at all. you can argue that the preprint does so "via the infinite series in an operation" (which I have no idea what that means; surely if exp(x) qualifies then so must 1/(x-y) if we pick a monomial basis?) but ¯\_(ツ)_/¯ now, do I think that this is groundbreaking magical research (as I'm currently seeing on twitter) no... But it's neat!