Arbitrary precision arithmetic using continued fractions

Functional languages supporting lazy evaluation invite novel applications, where conventional languages do not provide appropriate support for the problem.  In this paper we present an application of functional languages to arbitrary precision real arithmetic, using an unusual technique based on continued fractions, and depending crucially on lazy evaluation.

The precision of computer arithmetic calculations is normally decided by the programmer in advance, and is often hard to alter subsequently. Furthermore, without a formal analysis of the calculation, answers may be produced to spurious accuracy, and the computer gives no help in establishing error bounds for the result..  The technique presented here performs real arithmetic with guaranteed error bounds, in which the precision of the result can be arbitrarily increased without recommencing the calculation.

Here are two hard-to-find supporting publications

• Gosper’s Hackmem note on continued fractions
• Continued fractions without tears (Richards, 1981)