Abstract
The relative computability of the Cauchy function representation, the best approximation fractions, and the principal convergents of a real number is studied. It is proved that there exist real numbers x and y such that x has a polynomial-time computable Cauchy function but its best fractions are not polynomial-time computable, and that the best fractions of y are polynomial-time computable but its principal convergents are not polynomial-time computable.
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