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> Instead, the motivation he suggested was that you could run a computation in all the rounding modes, and if the results in all of them were reasonably close, you could be fairly certain that the result was accurate.

I actually thought that was the use case I was describing, though I would expect round-positive and round-negative to be enough. Don't the other rounding modes yield results within those bounds?

Consider something like

To obtain the correct lower bound, you need to round the addition up, and the subtraction down. This is also incredibly inefficient on current processors, as the registers are flushed after each mode change.

You're right. To be absolutely sure I tried writing a little program that does the thing I describe in the post, https://gist.github.com/plesner/7a13b5c8d269315df645, and sure enough it breaks down completely,

round default: 0.072817 round up: 0.072703 round down: 0.072931

Now I don't understand how you get a reliable result using rounding modes at all.

Interval arithmetic is tricky: you have to make sure you round in the most conservative direction for each operation. As other commenters mentioned, these bounds can end up being so large as to be useless.

The way I hear this is: you have two options.

- Either use interval arithmetic which is tricky and may give you useless large bounds, but those bounds are guaranteed to hold.

- Or just try all the rounding modes which is less likely to give you large bounds, but now you have no guarantee that those bounds say something meaningful about your computation.

Or does the second case give meaningful guarantees?

Not strictly, no. From what I understand, Kahan's idea was that you run the computation under different rounding modes and see how the behaviour changes, see for example:


Thanks for the link, I hadn't seen that. I have two reactions.

Firstly, I find his example contrived. He wants to determine the accuracy of a black box by subtly tweaking how floating-point operations spread throughout that black box behave? It seems like an okay hook for ad-hoc debugging -- if those flags are around anyway -- but as the primary rationale for the entire rounding mode mechanism? That's not a lot of bang for a whole lot of complexity.

Secondly, this is exactly the kind of discussion around rationales and use cases I'm looking for. Even if I don't buy his solution the problem still stands and might be solvable some other way, for instance through better control over external dependencies. Maybe something like newspeak's modules-as-objects and hierarchy inheritance mechanisms could be applied here.

I agree. I've never heard of anyone else utilising this approach, and allowing programs to change the mode of other programs is a recipe for shooting yourself in the foot.

One of the most infuriating things about global rounding modes is that they are so slow for things when changing rounding modes would actually be useful. A nice example is Euclidean division: given x,y, find the largest integer q such that y >= q*x (in exact arithmetic). If you change the rounding mode to down, then floor(y/x) would get you q. However the mode change is typically so slow that it is quicker to do something like round((y-fmod(y,x))/x).

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