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I worked on interval arithmetic and interval arithmetic-like alternatives to floating point (I was the first independent implementor of gustafson's "the end of error").

They get hairy really fast, and that's why ultimately when it came time to re-rethink floating point, we settled again on something with fixed bit with and inexact results.

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What do you mean, 'they get hairy really fast' ?

For starters, what happens to your numerical algorithm when it contains less-than comparisons? Intervals don't form an order.

It's not a total (heh) disaster, since many intervals are comparable with many other intervals. Floating point implicitly "solves" this problem by truncating the implicit interval onto a member of a fixed discrete set. It does so quite cleverly, but ultimately it still silently loses information; sometimes (admittedly rarely) you might want the correct result "I was unable to compute an answer due to lack of input precision" that happens when you try and work out whether [0,2] is less than [1,3].

Forget ordering. The useful equals relation becomes ternary.

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