Validated Constraint Compilation

We propose a method of using validated interval constraint contraction operators to build routines for numerical computation libraries. We illustrate the method by using it to construct a procedure that e ciently computes a small interval containing f 1(h) where f(x) = x sin(x). This can be transformed into a numerical routine by returning the midpoint of the interval and signalling an exception if the relative width of the interval exceeds a speci ed bound. We compare the interval method with a strictly numerical approach. The chief advantage of our method is that it results in a procedure which returns an interval that is guaranteed to contain the correct solution (assuming of course that the hardware and the compiler meet the necessary speci cations). By automating those parts of the computation which could e ect the correctness of the procedure, we reduce the number of places where the programmer can err. The user is free to design complex algorithms for solving the problem at hand, with the restriction that the algorithm consists of applying a sequence of validated constraint contractors which are constructed automatically from the constraint set specifying the problem. Modulo assumptions about the correctness of the contractors, the only error that the user can make is to produce a procedure which returns intervals that are too large too be useful. This is in contrast to traditional methods which always return answers with 52 bits of precision but with little or no indication of how many of those bits are correct.