On Sun, Dec 29, 2019 at 7:35 PM Andrew Barnert <abarnert@yahoo.com> wrote:
On Dec 29, 2019, at 15:19, David Mertz <mertz@gnosis.cx> wrote:On Sun, Dec 29, 2019, 5:20 PM Andrew Barnert via Python-ideas 
But it is, out of all of the possible magma-over-magma structures on those values, the one that most closely approximates—in a well-defined and useful, if very complicated, way—the rationals.

I'm sort of convinced that Posits better approximate the behavior of rationals than do IEEE-754 floats:


I played with whatever previous version of the concept is available in Julia, and it seemed cool. But I’m pretty sure they’re still not a field, and they don’t approximate the rationals (or any other field) more closely than IEEE floats do in the same well-defined way; rather, they’re the closest approximation in an equally well-defined but very different way, one that’s hopefully usually more useful in practice, but apparently even harder to define.

Of course posits are not a field.  

At a high level it's just a strategy to distribute deltas differently over the notional approximations that bit-patterns give of Rational numbers (not sure why Richard keeps insisting it's about Reals, not Rationals... albeit there is no difference that means anything to this discussion).  My understanding is that there is less "stair-stepping" in the size of an ULP in posits than in IEEE-754, basically; that seems roughly desirable, but not something I need to evangelize about.

I did have a really fun private conversation of zillions of emails with Uncle Timmy about this a few years back.  He can speak for himself, and knows way more than I do. But I think he's of a similar overall opinion.


--
Keeping medicines from the bloodstreams of the sick; food
from the bellies of the hungry; books from the hands of the
uneducated; technology from the underdeveloped; and putting
advocates of freedom in prisons.  Intellectual property is
to the 21st century what the slave trade was to the 16th.