Exact real arithmetic: a case study in higher order programming

Two methods for implementing ezact real arithmetic are explored One method is based on formulating real numbers as functions that map rational tolerances to rational approximations. This approach, which was developed by constructive mathematicians as a concrete formalization of the real numbers, has lead to a surprisingly successful implementation. The second method formulates real numbers as potentially infinite sequences of digits, evaluated on demand. This approach has frequently been advocated by proponents of lazy functional languages in the computer science community Ironically, it leads to much less satisfactory implementations. We discuss the theoretical problems involved m both methods, give algortthms for the basic arithmetic operations, and give an empirical comparison of the two techniques. We conclude wtth some general observations about the lazy evaluation paradigm and its implementation.