M000000317

$$\pi=\frac{33564865252904}{3}\arctan\left(\frac{1}{5727066010185}\right)+22117061740656\arctan\left(\frac{1}{5945958754193}\right)\frac{-96803421419680}{3}\arctan\left(\frac{1}{7601001102973}\right)+\frac{9600187052704}{3}\arctan\left(\frac{1}{7758917519457}\right)+\frac{50049039547028}{3}\arctan\left(\frac{1}{11244667034168}\right)\frac{-32432335337776}{3}\arctan\left(\frac{1}{13502703082318}\right)-27026931049832\arctan\left(\frac{1}{14483848717682}\right)+\frac{63062979014912}{3}\arctan\left(\frac{1}{15962244871097}\right)+\frac{45414359819648}{3}\arctan\left(\frac{1}{20293776947868}\right)+\frac{12454795941632}{3}\arctan\left(\frac{1}{33308851938318}\right)+14633561059808\arctan\left(\frac{1}{41887379804066}\right)\frac{-62561135901752}{3}\arctan\left(\frac{1}{69971515635443}\right)+2064033977772\arctan\left(\frac{1}{83979525486773}\right)\frac{-63631290930076}{3}\arctan\left(\frac{1}{127528611030443}\right)+7888837048592\arctan\left(\frac{1}{158682175700807}\right)+\frac{104200153739284}{3}\arctan\left(\frac{1}{187458398552950}\right)+\frac{36209491427644}{3}\arctan\left(\frac{1}{211773236092865}\right)\frac{-51641053160792}{3}\arctan\left(\frac{1}{240926005152903}\right)+\frac{17998961330884}{3}\arctan\left(\frac{1}{258213230677943}\right)+\frac{92124985812800}{3}\arctan\left(\frac{1}{317335434728073}\right)\frac{-23627086801492}{3}\arctan\left(\frac{1}{370392816588001}\right)+\frac{95658784555600}{3}\arctan\left(\frac{1}{758341235653128}\right)+\frac{42396423274888}{3}\arctan\left(\frac{1}{830202489183166}\right)+\frac{42276005058652}{3}\arctan\left(\frac{1}{1143900177872158}\right)+\frac{3747335841292}{3}\arctan\left(\frac{1}{1427056404016693}\right)\frac{-58404999379304}{3}\arctan\left(\frac{1}{2270451385422943}\right)+\frac{32474667159380}{3}\arctan\left(\frac{1}{4204987898062723}\right)+\frac{29860120605500}{3}\arctan\left(\frac{1}{6805164953551432}\right)$$

Compact formula	33564865252904/3[5727066010185] + 22117061740656[5945958754193] - 96803421419680/3[7601001102973] + 9600187052704/3[7758917519457] + 50049039547028/3[11244667034168] - 32432335337776/3[13502703082318] - 27026931049832[14483848717682] + 63062979014912/3[15962244871097] + 45414359819648/3[20293776947868] + 12454795941632/3[33308851938318] + 14633561059808[41887379804066] - 62561135901752/3[69971515635443] + 2064033977772[83979525486773] - 63631290930076/3[127528611030443] + 7888837048592[158682175700807] + 104200153739284/3[187458398552950] + 36209491427644/3[211773236092865] - 51641053160792/3[240926005152903] + 17998961330884/3[258213230677943] + 92124985812800/3[317335434728073] - 23627086801492/3[370392816588001] + 95658784555600/3[758341235653128] + 42396423274888/3[830202489183166] + 42276005058652/3[1143900177872158] + 3747335841292/3[1427056404016693] - 58404999379304/3[2270451385422943] + 32474667159380/3[4204987898062723] + 29860120605500/3[6805164953551432]
Lehmer's measure	1.99954
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 24 February 2004.] Download references as BibTeΧ