M000000355

$$\pi=-52925132666668\arctan\left(\frac{1}{10649932642403}\right)+37384055356972\arctan\left(\frac{1}{11244667034168}\right)-27274215251256\arctan\left(\frac{1}{13502703082318}\right)+74389393695648\arctan\left(\frac{1}{15962244871097}\right)+80254811744948\arctan\left(\frac{1}{20293776947868}\right)-8117949436676\arctan\left(\frac{1}{33308851938318}\right)-19709332116880\arctan\left(\frac{1}{41887379804066}\right)-97940028212092\arctan\left(\frac{1}{42512655792373}\right)+45974449618332\arctan\left(\frac{1}{53453286489686}\right)-28247663039596\arctan\left(\frac{1}{69971515635443}\right)+3145293644024\arctan\left(\frac{1}{83979525486773}\right)+24364467932152\arctan\left(\frac{1}{127528611030443}\right)+60957075381256\arctan\left(\frac{1}{131001068999312}\right)+69927172096100\arctan\left(\frac{1}{158682175700807}\right)-30201781076180\arctan\left(\frac{1}{187458398552950}\right)-25473125051708\arctan\left(\frac{1}{211773236092865}\right)-29405243525328\arctan\left(\frac{1}{240926005152903}\right)+119925578472740\arctan\left(\frac{1}{258213230677943}\right)-56603413886944\arctan\left(\frac{1}{317335434728073}\right)-65038433636464\arctan\left(\frac{1}{370392816588001}\right)-102515462554896\arctan\left(\frac{1}{758341235653128}\right)-30361497484296\arctan\left(\frac{1}{830202489183166}\right)+11973199001568\arctan\left(\frac{1}{1143900177872158}\right)+110377428576824\arctan\left(\frac{1}{1427056404016693}\right)-102207710348572\arctan\left(\frac{1}{2270451385422943}\right)+60593676016748\arctan\left(\frac{1}{4204987898062723}\right)+83551677443556\arctan\left(\frac{1}{4475336554336643}\right)-5275148387696\arctan\left(\frac{1}{5407789116841693}\right)+58046625838148\arctan\left(\frac{1}{6805164953551432}\right)-9113202380840\arctan\left(\frac{1}{79936438620894393}\right)$$

Compact formula	- 52925132666668[10649932642403] + 37384055356972[11244667034168] - 27274215251256[13502703082318] + 74389393695648[15962244871097] + 80254811744948[20293776947868] - 8117949436676[33308851938318] - 19709332116880[41887379804066] - 97940028212092[42512655792373] + 45974449618332[53453286489686] - 28247663039596[69971515635443] + 3145293644024[83979525486773] + 24364467932152[127528611030443] + 60957075381256[131001068999312] + 69927172096100[158682175700807] - 30201781076180[187458398552950] - 25473125051708[211773236092865] - 29405243525328[240926005152903] + 119925578472740[258213230677943] - 56603413886944[317335434728073] - 65038433636464[370392816588001] - 102515462554896[758341235653128] - 30361497484296[830202489183166] + 11973199001568[1143900177872158] + 110377428576824[1427056404016693] - 102207710348572[2270451385422943] + 60593676016748[4204987898062723] + 83551677443556[4475336554336643] - 5275148387696[5407789116841693] + 58046625838148[6805164953551432] - 9113202380840[79936438620894393]
Lehmer's measure	2.09211
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 May 2010.] Download references as BibTeΧ