> comparing sums of square roots is actually kind of a common subtask in eg computational geometry, so a bunch of their problems (including “shortest path through a graph in Euclidean space”!) are as-far-as-we-know extremely hard.

But the question then arises: do we always need to solve these problems to their fullest precision? Can we perhaps leave some ambiguity, and let the system gracefully deal with it? E.g. I don't care if my CAD model has a tiny chip of 1e-12 mm missing, as long as my CAD software doesn't crash on the resulting internal inconsistencies.

See also, e.g.: https://www.cs.purdue.edu/homes/cmh/distribution/papers/Robu...