Exact real arithmetic allows us to do computations without worrying about precision or rounding. In contrast with arbitrary precision arithmetic, we only need to specify an output precision and the details for intermediate steps are handled automatically.

In this talk we will implement exact real arithmetic in two very different ways. First, the "fast binary Cauchy" system amounts to representing each real as a function Natural -> Rational, such that each successive output is a closer approximation to the true value than the last. The second system represents each real number as a continued fraction; an infinite tower of sums and reciprocals. Both systems benefit from a functional programming style and the resulting code is very simple to understand.

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