Algorithm For Continued Fraction of $\pi$ without error.

Question

Is there an algorithm that will output the numbers in the continued fraction of $\pi$ without error? If one uses the usual method of calculating continued fractions, an approximation of $\pi$ is required, eventually causing errors. The algorithm should be able to work on computers with limited precision. The method would probably have to use the natural properties of $\pi$ that separate it from its approximations.