M000000357

$$\pi=12553278471988\arctan\left(\frac{1}{14483848717682}\right)+7562678842336\arctan\left(\frac{2}{34545983044739}\right)+6053467239208\arctan\left(\frac{1}{21881002771243}\right)+14093232553920\arctan\left(\frac{1}{22704779535001}\right)+5291484406596\arctan\left(\frac{1}{26573652775818}\right)-5017625761456\arctan\left(\frac{2}{64380110653261}\right)+5976373281420\arctan\left(\frac{1}{38057255532937}\right)+18590870332260\arctan\left(\frac{1}{41887379804066}\right)+19532309696520\arctan\left(\frac{2}{92253116774513}\right)-13614499404208\arctan\left(\frac{1}{55347780120443}\right)-521808714068\arctan\left(\frac{1}{69971515635443}\right)+9175543491716\arctan\left(\frac{1}{83979525486773}\right)-6982718524572\arctan\left(\frac{1}{127528611030443}\right)-5602177822476\arctan\left(\frac{1}{139889483331112}\right)+17881232043896\arctan\left(\frac{1}{187458398552950}\right)-2468139836400\arctan\left(\frac{1}{211773236092865}\right)+20291161062584\arctan\left(\frac{1}{317335434728073}\right)-21189354212332\arctan\left(\frac{1}{320714306992682}\right)-7605638686556\arctan\left(\frac{1}{370392816588001}\right)-9580080614636\arctan\left(\frac{1}{547264454742455}\right)+12316753859528\arctan\left(\frac{1}{758341235653128}\right)+11237643427276\arctan\left(\frac{1}{830202489183166}\right)+18719855499932\arctan\left(\frac{1}{977167012914932}\right)+18130568153896\arctan\left(\frac{1}{1143900177872158}\right)-1978394136784\arctan\left(\frac{1}{1427056404016693}\right)+4764378961120\arctan\left(\frac{1}{2270451385422943}\right)+13203724029160\arctan\left(\frac{1}{4204987898062723}\right)+7963656792112\arctan\left(\frac{1}{6604584685078343}\right)-16334309479728\arctan\left(\frac{1}{6805164953551432}\right)+1468710998180\arctan\left(\frac{1}{17251727577487177}\right)$$

Compact formula	12553278471988[14483848717682] + 7562678842336[34545983044739/2] + 6053467239208[21881002771243] + 14093232553920[22704779535001] + 5291484406596[26573652775818] - 5017625761456[64380110653261/2] + 5976373281420[38057255532937] + 18590870332260[41887379804066] + 19532309696520[92253116774513/2] - 13614499404208[55347780120443] - 521808714068[69971515635443] + 9175543491716[83979525486773] - 6982718524572[127528611030443] - 5602177822476[139889483331112] + 17881232043896[187458398552950] - 2468139836400[211773236092865] + 20291161062584[317335434728073] - 21189354212332[320714306992682] - 7605638686556[370392816588001] - 9580080614636[547264454742455] + 12316753859528[758341235653128] + 11237643427276[830202489183166] + 18719855499932[977167012914932] + 18130568153896[1143900177872158] - 1978394136784[1427056404016693] + 4764378961120[2270451385422943] + 13203724029160[4204987898062723] + 7963656792112[6604584685078343] - 16334309479728[6805164953551432] + 1468710998180[17251727577487177]
Lehmer's measure	2.09318
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 21 November 2009.] Download references as BibTeΧ