Optimizing programs over the constructive reals

The constructive reals provide programmers with a useful mechanism for prototyping numerical programs, and for experimenting with numerical algorithms. Unfortunately, the performance of current implementations is inadequate for some potential applications. In particular, these implementations tend to be space inefficient, in that they essentially require a complete computation history to be maintained.

Some numerical analysts (cf. [3]) propose that the programmer instead be provided with variable precision interval arithmetic, and then be required to write code to restart a computation when the intervals become too inaccurate. Though this model is no doubt appropriate at times, it is not an adequate replacement for exact arithmetic. The correct transformation from a program operating on the constructive reals to a reasonable program using iterated interval arithmetic can be nontrivial and error prone. Here we present a technique based on program slicing to both automate this process and reduce the amount of reexecution. Thus the programmer is still free to use the simpler abstraction of exact real arithmetic, but we can provide a more efficient interval arithmetic based implementation. Some preliminary empirical results are presented.