## 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.