# Distant decimals of $π$: Formal proofs of some algorithms computing them and guarantees of exact computation

1 MARELLE - Mathematical, Reasoning and Software
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We describe how to compute very far decimals of $π$ and how to provide formal guarantees that the decimals we compute are correct. In particular, we report on an experiment where 1 million decimals of $π$ and the billionth hexadecimal (without the preceding ones) have been computed in a formally verified way. Three methods have been studied, the first one relying on a spigot formula to obtain at a reasonable cost only one distant digit (more precisely a hexadecimal digit, because the numeration basis is 16) and the other two relying on arithmetic-geometric means. All proofs and computations can be made inside the Coq system. We detail the new formalized material that was necessary for this achievement and the techniques employed to guarantee the accuracy of the computed digits, in spite of the necessity to work with fixed precision numerical computation.
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Article dans une revue
Journal of Automated Reasoning, Springer Verlag, A Paraître, pp.1-45

Littérature citée [38 références]

https://hal.inria.fr/hal-01582524
Contributeur : Yves Bertot <>
Soumis le : lundi 11 décembre 2017 - 11:44:38
Dernière modification le : jeudi 11 janvier 2018 - 15:51:00

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• HAL Id : hal-01582524, version 2
• ARXIV : 1709.01743

### Citation

Yves Bertot, Laurence Rideau, Laurent Théry. Distant decimals of $π$: Formal proofs of some algorithms computing them and guarantees of exact computation. Journal of Automated Reasoning, Springer Verlag, A Paraître, pp.1-45. 〈hal-01582524v2〉

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