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## Section: New Results

### Properties of the $\pi$ number

Participants : Yves Bertot [correspondant] , Laurence Rideau, Laurent Théry.

As a testbed for the progress of formalized libraries in the domain of calculus, we studied an algorithm to compute $\pi$ (the circle ratio) using arithmetic-geometric means. This study brought us to extend the libraries with improper integrals, studies of arcsinh, variable change in integrals, and error propagation proofs.

We also studied a formal proof of the spigot algorithm designed by Bailey, Borwein, and Plouffe, which is used to compute far digits in the hexadecimal representation of $\pi$ as a fractional number. This relies on floating point computations and error control, for which we provided a formal proof.