Ask HN: Incremental Precision Solver

I have a system of equations or constraints over multiple variables. Most numerical solvers seem to compute some floating point solution at a fixed precision (float or double). I wonder if there are any approaches where the precision can be chosen, either upfront (user defined) or afterwards, that is, first a solution is given at some precision and then the user can ask for it to be improved (more iterations of whatever numerical solver), until user is happy with the precision. I assume perhaps interval arithmetic would be appropriate for this. Any pointers?