Distant decimals of $\pi$

TL;DR

This paper presents methods for computing extremely distant decimals of pi with formal correctness guarantees, including a verified computation of 1 million decimals and a billionth hexadecimal digit using Coq.

Contribution

It introduces verified algorithms for calculating distant pi decimals and reports on the first formal proof of such a large-scale computation within Coq.

Findings

Successfully computed 1 million decimals of pi with formal verification.

Achieved a billionth hexadecimal digit of pi in a verified manner.

Developed new formalized techniques for fixed precision numerical computations.

Abstract

We describe how to compute very far decimals of $$\pi$$ 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 $$\pi$$ 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.