The interval based argument reduction presented in the last subsection will correctly handle infinite arguments without table lookup. Even with limited understanding of the method used to compute , both and may be implemented. An implementation of would return u . The implementation may be extended, as the first was, to allow and to be computed for arbitrary arguments. Although the two methods initially appear to be distinct, the first implementation of is simply a cleverly optimized version of the second implementation of .
Further optimization is possible.
In general, interval methods allow one to build and from and , using the method of computing from as a guide. Knowledge of and can help produce good implementations of and .