[dahart]

> That's autodiff! :D The error (change) in a function's output given an error (change) in its input is the definition of its derivative

Ah but the error of your autodiff math isn’t the same thing as the error of the function you’re autodiffing! And the math error isn’t even a derivative. Specifically, the absolute value handling of rounding error analysis will cause the error bounds to diverge from being proportional to the function and/or derivative values.

I assume your unknown sum sizes are eventually finite… otherwise you won’t get an answer? Evaluating the rounding error will just take the same time as evaluating your sum. Long sums, btw, are bad for precision. Typically people try to sort them or compute them in hierarchical/ butterfly reduce fashion, or use other tricks.

[tomsmeding]

> Specifically, the absolute value handling of rounding error analysis will cause the error bounds to diverge from being proportional to the function and/or derivative values.

Oh, I see! Not quite autodiff then, indeed.

> Long sums, btw, are bad for precision. Typically people try to sort them or compute them in hierarchical/ butterfly reduce fashion, or use other tricks.

Typical parallel reduction algorithms do precisely this, but for different reasons (parallelisation). I guess I'd have to compute a partially symbolic expression for the error bound depending on the array size, and hope that I'm not going to change the parallel reduction algorithm without changing the error bounds in tests.

It's possible, but finicky. I see why people slap simple relative bounds on things and call it a day. :)

[dahart]

There’s probably a way to get a stable conservative error bound for a parallel sum reduction that’s non-deterministic.

Here’s the PBR book’s chapter on error analysis. It’s way easier to implement error bound tracking than it is to come up with symbolic bounds, and it’s way better than having nothing or even a guess pulled out of thin air. It’s probably worth trying, just to see what happens, it might be surprising. Plus if you implement the error bounds, you may find very interesting and/or enlightening ways to visualize them.

https://pbr-book.org/4ed/Shapes/Managing_Rounding_Error#

[tomsmeding]

Thanks a lot for the insights!