Re: Unbounded intervals
On 2012-04-25 10:11:58 -0700, Dan Zuras Intervals wrote:
> We had no difficulty with 1/Small = +inf, Exp(Huge) = +inf,
> Huge^2 = +inf, or Huge + Huge = +inf or any of the more
> subtle distinctions that are being made with OVR versus
> +inf.
>
> Indeed, this entire discussion would not have taken place
> within the 754 world. It would have been as meaningless
> there as I believe it will turn out to be here, in the end.
>
> In the end, we will all have to have some symbol Big to
> stand for some number beyond our ability to calculate with
> any degree of accuracy. We will have to have, 1/Small = Big,
> Exp(Huge) = Big, Huge^2 = Big, & Huge + Huge = Big for all
> the Small(s) & Huge(s) we can calculate with accurately.
> Whether we call Big by the name OVR or +Inf will not matter
> to us.
There are differences between IEEE 754 and P1788:
* IEEE 754 covers approximate computations (one can also do exact
computations, but this is not discussed here). Thanks to correct
rounding, one can compute error bounds and do proofs. Such bounds
are typically incorrect when an overflow or underflow occurs, but
one can check the exception flags. Concerning the infinities,
they are not real numbers, and if they are added at Level 1,
things may become inconsistent if one goes too far, like what
has been done (for instance, with 1/+0 vs 1/-0).
* In P1788, approximate computations also occur, but more
importantly, we want to require the containment property,
so that results are valid (the specific case of partially
undefined functions can be handled via decorations). Unbounded
intervals are part of the math theory (which corresponds to
Level 1), e.g.
http://en.wikipedia.org/wiki/Interval_%28mathematics%29
http://mathworld.wolfram.com/Interval.html
or any math textbook. This is *not* something artificial like
what has partly been done with the infinities in IEEE 754. So,
there are no good reasons to reject unbounded intervals.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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