M000000345

$$\pi=21129264168664\arctan\left(\frac{2}{17183229925989}\right)+6823574864300\arctan\left(\frac{1}{10546511281730}\right)-16504518845836\arctan\left(\frac{1}{11244667034168}\right)+5844842165004\arctan\left(\frac{2}{23780027088431}\right)+14379952280252\arctan\left(\frac{1}{13502703082318}\right)-5410831807000\arctan\left(\frac{2}{28456536638489}\right)+19818592306560\arctan\left(\frac{1}{15420002333757}\right)-15059692096284\arctan\left(\frac{1}{15962244871097}\right)+921107691140\arctan\left(\frac{2}{33377023452861}\right)+8393894999540\arctan\left(\frac{1}{24405694099301}\right)+7202047394120\arctan\left(\frac{1}{33308851938318}\right)-13647694603516\arctan\left(\frac{1}{41887379804066}\right)-6673961260308\arctan\left(\frac{1}{42859110883727}\right)+8987022231700\arctan\left(\frac{1}{69971515635443}\right)-10161797458260\arctan\left(\frac{1}{74230400727943}\right)-522104260064\arctan\left(\frac{1}{94465125845354}\right)-15759613218088\arctan\left(\frac{1}{158682175700807}\right)-3887588006700\arctan\left(\frac{1}{187458398552950}\right)+6414127064184\arctan\left(\frac{1}{211773236092865}\right)+6318330552492\arctan\left(\frac{1}{240926005152903}\right)-6460508072084\arctan\left(\frac{1}{258213230677943}\right)-8074313637328\arctan\left(\frac{1}{320714306992682}\right)-5596740940736\arctan\left(\frac{1}{370392816588001}\right)-7856411406132\arctan\left(\frac{1}{758341235653128}\right)+546419774916\arctan\left(\frac{1}{830202489183166}\right)+16925439233484\arctan\left(\frac{1}{1421149505569643}\right)-13622959061536\arctan\left(\frac{1}{1427056404016693}\right)+4961569265516\arctan\left(\frac{1}{2270451385422943}\right)-3392123195616\arctan\left(\frac{1}{4204987898062723}\right)$$

Compact formula	21129264168664[17183229925989/2] + 6823574864300[10546511281730] - 16504518845836[11244667034168] + 5844842165004[23780027088431/2] + 14379952280252[13502703082318] - 5410831807000[28456536638489/2] + 19818592306560[15420002333757] - 15059692096284[15962244871097] + 921107691140[33377023452861/2] + 8393894999540[24405694099301] + 7202047394120[33308851938318] - 13647694603516[41887379804066] - 6673961260308[42859110883727] + 8987022231700[69971515635443] - 10161797458260[74230400727943] - 522104260064[94465125845354] - 15759613218088[158682175700807] - 3887588006700[187458398552950] + 6414127064184[211773236092865] + 6318330552492[240926005152903] - 6460508072084[258213230677943] - 8074313637328[320714306992682] - 5596740940736[370392816588001] - 7856411406132[758341235653128] + 546419774916[830202489183166] + 16925439233484[1421149505569643] - 13622959061536[1427056404016693] + 4961569265516[2270451385422943] - 3392123195616[4204987898062723]
Lehmer's measure	2.08000
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 04 August 2006.] Download references as BibTeΧ