The Chudnovsky algorithm computes Pi to high precision by evaluating the Chudnovsky series with controlled truncation and rounding errors. A direct implementation updates the ratio of successive terms rather than recomputing every factorial. Large computations use binary splitting to combine rational partial sums through a balanced recursion tree, reducing expensive high-precision divisions. Roughly one term is needed per fourteen requested decimal digits, with additional guard precision and a bound on the remaining tail. Fast convergence of the series does not alone certify printed digits: finite-precision arithmetic errors must also be bounded.
Binary splitting evaluates a finite rational sum or a hypergeometric series by recursively dividing its term range into two nearly equal parts and combining exact integer data. The resulting balanced recursion tree postpones high-precision divisions and can exploit fast multiplication of similarly sized integers.
For ordinary rational partial sums represented by and , the exact merge isFor example, merging and yields . Hypergeometric applications additionally carry products of successive-term ratio numerators and denominators to avoid rebuilding them at every node.
In the Chudnovsky algorithm, this organization makes large integer arithmetic efficient while evaluating the rational partial sum exactly before the final conversion and square-root factors. It does not bound the omitted infinite tail or floating-point rounding by itself: those need separate error estimates. The benefit is the balanced arithmetic organization, not a change in the value of the series.
The Chudnovsky series is the rapidly converging identityThe absolute ratio of successive summands tends to , approximately , giving about fourteen new decimal digits per term asymptotically. Keeping only gives . The factorial ratio, not a direct geometric approximation of a circle, controls the numerical convergence.[ref]
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