4 ms·

Unless you allow the denominator to grow unbound, rational have no real advantage over FP and are much slower. If you allow unbound denominator, you will run ou

Unless you allow the denominator to grow unbound, rational have no real advantage over FP and are much slower. If you allow unbound denominator, you will run out of memory after perhaps a few thousand operations (and having to dynamically allocate memory will kill performance even further). And then there are transcendent functions which cannot be represented by rationals anyway.

crabbone 2y ago
> And then there are transcendent functions which cannot be represented by rationals anyway. They cannot be represented by floats either. So this doesn't matter.
- dzaima 2y ago
  The meaningful difference with irrational computations on rationals vs floats is that in floats you have some inherent, expected, and well-understood imprecision that applies evenly over results of add/mul/div/sqrt/sin/etc, whereas in unbounded rationals add/mul/div are all infinitely precise always (and presumably this property is why one would want rationals over floats) but has no such possible equivalent for sqrt/sin (at which point your options are either just giving up, or capping the required precision, at which point you could just use floats).
  - wakawaka28 2y ago
    Floating point operations also run in constant-bounded time. You can't say that about any arbitrary-precision format.
  - crabbone 2y ago
    > well-understood imprecision that applies evenly over results I'm sorry, I have some bad news for you... especially the "evenly" part. I had a friend who worked at IBM research on the problem of "normalizing" some matrix multiplication irregularities which occur due to numerical underflow or overflow. And, it's anything but a solved or an easy problem.
    - dzaima 2y ago
      There a maximum error of 0.5 ulp for all of add/sub/mul/div/sqrt on all inputs (ok sin/cos & co typically have a couple ulp of error but close enough). And indeed it's well-understood that it can get problematic if you compose a bunch of such, esp. add/sub with a large magnitude difference. Indeed, especially matrix multiplication.

crabbone 2y ago

> And then there are transcendent functions which cannot be represented by rationals anyway. They cannot be represented by floats either. So this doesn't matter.

dzaima 2y ago
The meaningful difference with irrational computations on rationals vs floats is that in floats you have some inherent, expected, and well-understood imprecision that applies evenly over results of add/mul/div/sqrt/sin/etc, whereas in unbounded rationals add/mul/div are all infinitely precise always (and presumably this property is why one would want rationals over floats) but has no such possible equivalent for sqrt/sin (at which point your options are either just giving up, or capping the required precision, at which point you could just use floats).
- wakawaka28 2y ago
  Floating point operations also run in constant-bounded time. You can't say that about any arbitrary-precision format.
- crabbone 2y ago
  > well-understood imprecision that applies evenly over results I'm sorry, I have some bad news for you... especially the "evenly" part. I had a friend who worked at IBM research on the problem of "normalizing" some matrix multiplication irregularities which occur due to numerical underflow or overflow. And, it's anything but a solved or an easy problem.
  - dzaima 2y ago
    There a maximum error of 0.5 ulp for all of add/sub/mul/div/sqrt on all inputs (ok sin/cos & co typically have a couple ulp of error but close enough). And indeed it's well-understood that it can get problematic if you compose a bunch of such, esp. add/sub with a large magnitude difference. Indeed, especially matrix multiplication.