2017/09/06 by Yves Bertot, Bertot, Yves, Laurence Rideau +3
Computer Science · Mathematics · #Advanced Algebra and Geometry #Coding theory and cryptography #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #Logic in Computer Science (cs.LO) #Logic, programming, and type systems #Mathematical Approximation and Integration #Numerical Methods and Algorithms
paper · pdf · doi:10.48550/arxiv.1709.01743
openalex publication_date 2017/09/06 · openalex created_date 2019/12/26 · openalex updated_date 2026/07/28
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.