M000000354

$$\pi=6638759392758\arctan\left(\frac{1}{14483848717682}\right)-11768347131013\arctan\left(\frac{1}{15962244871097}\right)+\frac{66139371352551}{2}\arctan\left(\frac{1}{16083127751238}\right)\frac{-51286923607451}{2}\arctan\left(\frac{1}{16248096944867}\right)+15297959132142\arctan\left(\frac{1}{16326085782557}\right)+\frac{36498915372523}{2}\arctan\left(\frac{1}{19507778531302}\right)+13618567891689\arctan\left(\frac{1}{20293776947868}\right)+25813104752086\arctan\left(\frac{1}{21602138452318}\right)+3937544159823\arctan\left(\frac{2}{69569213044365}\right)\frac{-41580777192839}{2}\arctan\left(\frac{1}{36918949984093}\right)-11151289954787\arctan\left(\frac{1}{69971515635443}\right)\frac{-44427687575869}{2}\arctan\left(\frac{1}{83979525486773}\right)+\frac{25961834284111}{2}\arctan\left(\frac{2}{194151845499011}\right)+\frac{30168587862015}{2}\arctan\left(\frac{1}{129271780546462}\right)\frac{-74409661277039}{2}\arctan\left(\frac{1}{158682175700807}\right)+\frac{9673346827427}{2}\arctan\left(\frac{1}{256093522057693}\right)+7237358237595\arctan\left(\frac{1}{309087578314273}\right)-2386183523636\arctan\left(\frac{1}{474397757071282}\right)+\frac{189495491757}{2}\arctan\left(\frac{1}{600662758381590}\right)\frac{-10264867439083}{2}\arctan\left(\frac{1}{686308367425978}\right)+627179674429\arctan\left(\frac{1}{832181505450757}\right)+\frac{1070813886195}{2}\arctan\left(\frac{1}{1367017926441055}\right)+\frac{45416794855829}{2}\arctan\left(\frac{1}{1443713070245525}\right)+\frac{57396108038221}{2}\arctan\left(\frac{1}{2799978903689557}\right)+13045313450007\arctan\left(\frac{2}{7783134400901473}\right)+10162424081001\arctan\left(\frac{1}{4088593248636842}\right)+\frac{8073515547771}{2}\arctan\left(\frac{1}{4161072789242257}\right)\frac{-7630931865843}{2}\arctan\left(\frac{1}{6805164953551432}\right)\frac{-23436369811495}{2}\arctan\left(\frac{1}{7474541196075273}\right)+12378302157923\arctan\left(\frac{1}{74295753693510382}\right)$$

Compact formula	6638759392758[14483848717682] - 11768347131013[15962244871097] + 66139371352551/2[16083127751238] - 51286923607451/2[16248096944867] + 15297959132142[16326085782557] + 36498915372523/2[19507778531302] + 13618567891689[20293776947868] + 25813104752086[21602138452318] + 3937544159823[69569213044365/2] - 41580777192839/2[36918949984093] - 11151289954787[69971515635443] - 44427687575869/2[83979525486773] + 25961834284111/2[194151845499011/2] + 30168587862015/2[129271780546462] - 74409661277039/2[158682175700807] + 9673346827427/2[256093522057693] + 7237358237595[309087578314273] - 2386183523636[474397757071282] + 189495491757/2[600662758381590] - 10264867439083/2[686308367425978] + 627179674429[832181505450757] + 1070813886195/2[1367017926441055] + 45416794855829/2[1443713070245525] + 57396108038221/2[2799978903689557] + 13045313450007[7783134400901473/2] + 10162424081001[4088593248636842] + 8073515547771/2[4161072789242257] - 7630931865843/2[6805164953551432] - 23436369811495/2[7474541196075273] + 12378302157923[74295753693510382]
Lehmer's measure	2.09165
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