Very cool. This feels like one of those ideas that makes interval arithmetic go from “interesting but frustrating” to actually useful. I’d be curious how you handle the growth in the number of disjoint intervals over repeated operations, since that seems like the practical bottleneck.

I don't handle it, ahah. You are right that if you take any classical numerical computing algorithm and replace the floating point reals by interval unions, most of the time the number of intervals in the unions in each of your variables will grow very fast. This is one of the problems of unions and as far as I'm aware it's a topic of active academic research.