M000000348

$$\pi=8109980481740\arctan\left(\frac{1}{16083127751238}\right)-1189026102180\arctan\left(\frac{1}{16248096944867}\right)-2224802852104\arctan\left(\frac{1}{16326085782557}\right)+3168114551516\arctan\left(\frac{1}{19337029786167}\right)+28218176030212\arctan\left(\frac{1}{19507778531302}\right)+18245343328936\arctan\left(\frac{1}{20293776947868}\right)-13539520253584\arctan\left(\frac{1}{21602138452318}\right)-10722255987816\arctan\left(\frac{2}{69569213044365}\right)+6381417331708\arctan\left(\frac{1}{36918949984093}\right)+13541534903244\arctan\left(\frac{1}{69971515635443}\right)+35958359688036\arctan\left(\frac{1}{79507238248932}\right)+10207107500436\arctan\left(\frac{1}{93544794530637}\right)+20058182469380\arctan\left(\frac{2}{194151845499011}\right)+42401691469160\arctan\left(\frac{1}{129271780546462}\right)-14859916925720\arctan\left(\frac{2}{269506107350815}\right)-9634542643636\arctan\left(\frac{1}{309087578314273}\right)+228607958100\arctan\left(\frac{1}{412252392763333}\right)-7487787074436\arctan\left(\frac{1}{600662758381590}\right)-4894044563056\arctan\left(\frac{1}{686308367425978}\right)-15913534503856\arctan\left(\frac{1}{832181505450757}\right)+28814330802288\arctan\left(\frac{1}{1015679391413874}\right)-16098104781648\arctan\left(\frac{1}{1367017926441055}\right)-23721492905080\arctan\left(\frac{1}{1443713070245525}\right)-20954476033456\arctan\left(\frac{1}{2799978903689557}\right)-35863557953084\arctan\left(\frac{2}{7783134400901473}\right)+9857271366092\arctan\left(\frac{1}{4088593248636842}\right)+24038500828228\arctan\left(\frac{1}{4161072789242257}\right)-15996285010464\arctan\left(\frac{1}{6805164953551432}\right)+3656782711348\arctan\left(\frac{1}{7474541196075273}\right)-12314522700108\arctan\left(\frac{1}{74295753693510382}\right)$$

Compact formula	8109980481740[16083127751238] - 1189026102180[16248096944867] - 2224802852104[16326085782557] + 3168114551516[19337029786167] + 28218176030212[19507778531302] + 18245343328936[20293776947868] - 13539520253584[21602138452318] - 10722255987816[69569213044365/2] + 6381417331708[36918949984093] + 13541534903244[69971515635443] + 35958359688036[79507238248932] + 10207107500436[93544794530637] + 20058182469380[194151845499011/2] + 42401691469160[129271780546462] - 14859916925720[269506107350815/2] - 9634542643636[309087578314273] + 228607958100[412252392763333] - 7487787074436[600662758381590] - 4894044563056[686308367425978] - 15913534503856[832181505450757] + 28814330802288[1015679391413874] - 16098104781648[1367017926441055] - 23721492905080[1443713070245525] - 20954476033456[2799978903689557] - 35863557953084[7783134400901473/2] + 9857271366092[4088593248636842] + 24038500828228[4161072789242257] - 15996285010464[6805164953551432] + 3656782711348[7474541196075273] - 12314522700108[74295753693510382]
Lehmer's measure	2.08476
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