@coudy I wish I would also show another formula for $\pi$ derived by using the Borwein integrals involving sinc function as shown in the paper by Uwe Bäsel and Robert Baillie: arxiv.org/pdf/1510.03200.pdf (published in ELEMENTE DER MATHEMATIK, Volume 71, Issue 1, 2016, pp. 7–20 doi.org/10.4171/EM/295). Unfortunately the window is a bit narrow here to fit a Bäsel–Baillie formula for $\pi$ requiring a ratio with $453\,130\,145$ and $453\,237\,170$ digits in numerator and denominator.