Not an answer to your question, but I found out recently that there is definitely significant interest in ideas like interval arithmetic in hardware among mathematicians. The application I am somewhat familiar with is "brute forcing" proofs of inequalities, which can get very annoying, even for apparently easy stuff like 2 variable ones.
Indeed, "The Princeton Companion to Applied Mathematics" edited by N. Higham et al has an entire article on this (in section II.20). In particular, the article has a reference to a library called filib++ (http://www2.math.uni-wuppertal.de/~xsc/software/filib.html). I unfortunately have no idea how it performs/how good it is.
By the way, thanks a lot for the Unum reference; seems like it is still in too early a stage for inclusion in the book above.