M000000358

$$\pi=\frac{-508117940916613}{6}\arctan\left(\frac{1}{13420523969127}\right)+\frac{347289466481843}{3}\arctan\left(\frac{1}{14483848717682}\right)\frac{-405141821660555}{6}\arctan\left(\frac{1}{15962244871097}\right)-6521328485880\arctan\left(\frac{1}{16083127751238}\right)+\frac{37718108813950}{3}\arctan\left(\frac{1}{16248096944867}\right)+1176154650409\arctan\left(\frac{1}{16326085782557}\right)\frac{-307840992948641}{6}\arctan\left(\frac{1}{18202617562057}\right)+71093093961298\arctan\left(\frac{1}{19507778531302}\right)-32002211445998\arctan\left(\frac{1}{20293776947868}\right)+\frac{261763342154756}{3}\arctan\left(\frac{1}{21602138452318}\right)+\frac{335590806168919}{6}\arctan\left(\frac{2}{69569213044365}\right)\frac{-132046699462630}{3}\arctan\left(\frac{1}{36918949984093}\right)+\frac{117815123995975}{2}\arctan\left(\frac{1}{69971515635443}\right)+\frac{157520053302493}{6}\arctan\left(\frac{2}{194151845499011}\right)+50226850438857\arctan\left(\frac{1}{129271780546462}\right)\frac{-23118329922737}{2}\arctan\left(\frac{1}{309087578314273}\right)\frac{-31082314612951}{6}\arctan\left(\frac{1}{392998900984198}\right)\frac{-101632261639385}{6}\arctan\left(\frac{1}{600662758381590}\right)\frac{-35548870815889}{3}\arctan\left(\frac{1}{686308367425978}\right)+\frac{693716317528003}{6}\arctan\left(\frac{1}{832181505450757}\right)+\frac{282033394124945}{6}\arctan\left(\frac{1}{1367017926441055}\right)\frac{-73744463390045}{2}\arctan\left(\frac{1}{1443713070245525}\right)\frac{-2891748757085}{2}\arctan\left(\frac{1}{2799978903689557}\right)\frac{-183002497529324}{3}\arctan\left(\frac{2}{7783134400901473}\right)\frac{-188826367498073}{6}\arctan\left(\frac{1}{4088593248636842}\right)+\frac{288724397967857}{6}\arctan\left(\frac{1}{4161072789242257}\right)\frac{-132252094283093}{6}\arctan\left(\frac{1}{6805164953551432}\right)+\frac{467919417916181}{6}\arctan\left(\frac{1}{7474541196075273}\right)\frac{-138379339167845}{6}\arctan\left(\frac{1}{19788936733439557}\right)\frac{-188582806691030}{3}\arctan\left(\frac{1}{74295753693510382}\right)$$

Compact formula	- 508117940916613/6[13420523969127] + 347289466481843/3[14483848717682] - 405141821660555/6[15962244871097] - 6521328485880[16083127751238] + 37718108813950/3[16248096944867] + 1176154650409[16326085782557] - 307840992948641/6[18202617562057] + 71093093961298[19507778531302] - 32002211445998[20293776947868] + 261763342154756/3[21602138452318] + 335590806168919/6[69569213044365/2] - 132046699462630/3[36918949984093] + 117815123995975/2[69971515635443] + 157520053302493/6[194151845499011/2] + 50226850438857[129271780546462] - 23118329922737/2[309087578314273] - 31082314612951/6[392998900984198] - 101632261639385/6[600662758381590] - 35548870815889/3[686308367425978] + 693716317528003/6[832181505450757] + 282033394124945/6[1367017926441055] - 73744463390045/2[1443713070245525] - 2891748757085/2[2799978903689557] - 183002497529324/3[7783134400901473/2] - 188826367498073/6[4088593248636842] + 288724397967857/6[4161072789242257] - 132252094283093/6[6805164953551432] + 467919417916181/6[7474541196075273] - 138379339167845/6[19788936733439557] - 188582806691030/3[74295753693510382]
Lehmer's measure	2.09334
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