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Exact Real Computer Arithmetic with Continued Fractions
August 1990 (vol. 39 no. 8)
pp. 1087-1105

A representation of the computable real numbers by continued fractions is introduced. This representation deals with the subtle points of undecidable comparison and integer division, as well as representing the infinite 1/0 and undefined 0/0 numbers. Two general algorithms for performing arithmetic operations are introduced. The algebraic algorithm, which computes sums and products of continued fractions as a special case, basically operates in a positional manner, producing one term of output for each term of input. The transcendental algorithm uses a general formula of Gauss to compute the continued fractions of exponentials, logarithms, trigonometric functions, and a wide class of special functions. A prototype system has been implemented in LeLisp and the performance of these algorithms is promising.

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Index Terms:
exact real computer arithmetic; continued fractions; computable real numbers; undecidable comparison; integer division; infinite 1/0; undefined 0/0 numbers; arithmetic operations; algebraic algorithm; sums; products; positional; transcendental algorithm; Gauss; exponentials; logarithms; trigonometric functions; special functions; LeLisp; digital arithmetic; number theory.
J.E. Vuillemin, "Exact Real Computer Arithmetic with Continued Fractions," IEEE Transactions on Computers, vol. 39, no. 8, pp. 1087-1105, Aug. 1990, doi:10.1109/12.57047
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