The problem considered is that of evaluating a rational expression to within any desired tolerance on a computer which performs variable-precision floating-point arithmetic operations. For example, the expression might be π/(π + 1/2 - e) √2), which is rational in the data π, e, √2. An automatic error analysis technique is given for determining, directly from the results of a trial low-precision interval arithmetic calculation, just how much precision and data accuracy are required to achieve a desired final accuracy. The techniques given generalize easily to the evaluation of many nonrational expressions.