M000000349

$$\pi=4396599624182\arctan\left(\frac{1}{14483848717682}\right)\frac{-44565151799231}{2}\arctan\left(\frac{1}{15962244871097}\right)+20529188715768\arctan\left(\frac{1}{16083127751238}\right)\frac{-15267479219661}{2}\arctan\left(\frac{1}{16248096944867}\right)+8987851957579\arctan\left(\frac{1}{16326085782557}\right)+21965051215687\arctan\left(\frac{1}{19507778531302}\right)+8573564356474\arctan\left(\frac{1}{20293776947868}\right)+\frac{8463625567651}{2}\arctan\left(\frac{1}{21602138452318}\right)+17534814638177\arctan\left(\frac{2}{69569213044365}\right)+\frac{6482786523837}{2}\arctan\left(\frac{1}{36918949984093}\right)+\frac{6285677419365}{2}\arctan\left(\frac{1}{69971515635443}\right)+\frac{18786297768629}{2}\arctan\left(\frac{2}{175614682377261}\right)+\frac{27226937923459}{2}\arctan\left(\frac{2}{194151845499011}\right)+\frac{80762358857387}{2}\arctan\left(\frac{1}{129271780546462}\right)-6166083419503\arctan\left(\frac{1}{169838669284032}\right)\frac{-43550241579033}{2}\arctan\left(\frac{1}{309087578314273}\right)-1234662259911\arctan\left(\frac{1}{600662758381590}\right)-8848027248967\arctan\left(\frac{1}{686308367425978}\right)+\frac{54506950355809}{2}\arctan\left(\frac{1}{832181505450757}\right)+\frac{35314736634667}{2}\arctan\left(\frac{1}{963777344919407}\right)+4251000472523\arctan\left(\frac{1}{1367017926441055}\right)\frac{-692656602391}{2}\arctan\left(\frac{1}{1443713070245525}\right)+5090493858945\arctan\left(\frac{1}{2799978903689557}\right)+20884433782145\arctan\left(\frac{1}{3386696477716649}\right)\frac{-39494130945943}{2}\arctan\left(\frac{2}{7783134400901473}\right)+\frac{16688972021195}{2}\arctan\left(\frac{1}{4088593248636842}\right)-799915639828\arctan\left(\frac{1}{4161072789242257}\right)-7531059462347\arctan\left(\frac{1}{6805164953551432}\right)+3208495578396\arctan\left(\frac{1}{7474541196075273}\right)\frac{-28495986691105}{2}\arctan\left(\frac{1}{74295753693510382}\right)$$

Compact formula	4396599624182[14483848717682] - 44565151799231/2[15962244871097] + 20529188715768[16083127751238] - 15267479219661/2[16248096944867] + 8987851957579[16326085782557] + 21965051215687[19507778531302] + 8573564356474[20293776947868] + 8463625567651/2[21602138452318] + 17534814638177[69569213044365/2] + 6482786523837/2[36918949984093] + 6285677419365/2[69971515635443] + 18786297768629/2[175614682377261/2] + 27226937923459/2[194151845499011/2] + 80762358857387/2[129271780546462] - 6166083419503[169838669284032] - 43550241579033/2[309087578314273] - 1234662259911[600662758381590] - 8848027248967[686308367425978] + 54506950355809/2[832181505450757] + 35314736634667/2[963777344919407] + 4251000472523[1367017926441055] - 692656602391/2[1443713070245525] + 5090493858945[2799978903689557] + 20884433782145[3386696477716649] - 39494130945943/2[7783134400901473/2] + 16688972021195/2[4088593248636842] - 799915639828[4161072789242257] - 7531059462347[6805164953551432] + 3208495578396[7474541196075273] - 28495986691105/2[74295753693510382]
Lehmer's measure	2.08499
References	Wetherfield, Michael Roby and Chien-Lih, Hwang. Computing pi: Lists of Machin-type (inverse cotangent) identities for pi/4, [Online; accessed 04-February-2024], 2013. [web.archive.org/web/20240204042153/http] [Found by MRW on 19 August 2011.] Download references as BibTeΧ