Exactly solving rst-order constraints (i.e., rst-order formulas over a certain prede ned structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers it is promising to compute approximate solutions instead of exact ones. However, the quanti ers of the rst-order predicate language are an obstacle to allowing approximations to arbitrary small error bounds. In this paper we remove this obstacle by modifying the rst-order language and replacing the classical quanti ers with approximate quanti ers. These also have two additional advantages: First, they are tunable, in the sense that they allow the user to decide on the trade-o between precision and e ciency. Second, they introduce additional expressivity into the rst-order language by allowing reasoning over the size of solution sets.