  Exactly solving first-order constraints (i.e., first-order formulas over a
certain predefined 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 quantifiers of the
first-order predicate language are an obstacle to allowing approximations to
arbitrary small error bounds. In this paper we solve the problem by modifying
the first-order language and replacing the classical quantifiers with
approximate quantifiers. These also have two additional advantages: First, they
are tunable, in the sense that they allow the user to decide on the trade-off
between precision and efficiency. Second, they introduce additional
expressivity into the first-order language by allowing reasoning over the size
of solution sets.
