M000000331

$$\pi=1123486720403\arctan\left(\frac{1}{1259851011582}\right)+1745057227773\arctan\left(\frac{1}{1686145264193}\right)+1430433470707\arctan\left(\frac{1}{1921044951457}\right)-1027813353382\arctan\left(\frac{1}{2401808546428}\right)-993823298739\arctan\left(\frac{2}{5345202735279}\right)-1053919165144\arctan\left(\frac{1}{3486467472132}\right)+463817352739\arctan\left(\frac{1}{3621667588383}\right)+1999905303948\arctan\left(\frac{1}{5142102426318}\right)+1368187568227\arctan\left(\frac{1}{5602034310634}\right)+2378564121418\arctan\left(\frac{1}{5688475754312}\right)-783006883663\arctan\left(\frac{1}{6202471697443}\right)+1415985932436\arctan\left(\frac{1}{7580898173568}\right)+2596112296801\arctan\left(\frac{1}{15962244871097}\right)-121109185327\arctan\left(\frac{1}{18104444568932}\right)+2538113946637\arctan\left(\frac{1}{18203022356332}\right)+233665967783\arctan\left(\frac{1}{23064526452057}\right)+838739800829\arctan\left(\frac{2}{74963671338441}\right)-562789014113\arctan\left(\frac{1}{38057255532937}\right)+1691756801872\arctan\left(\frac{1}{69971515635443}\right)-2492641396317\arctan\left(\frac{1}{180481359172943}\right)+1540241134881\arctan\left(\frac{1}{240926005152903}\right)+536745742237\arctan\left(\frac{1}{258703261539143}\right)+38727830367\arctan\left(\frac{1}{389928169608307}\right)-517173422907\arctan\left(\frac{1}{686308367425978}\right)+1768427532407\arctan\left(\frac{1}{2549868113472987}\right)+1557551763814\arctan\left(\frac{1}{4439568777547493}\right)+1044205452522\arctan\left(\frac{1}{8489727499760268}\right)$$

Compact formula	1123486720403[1259851011582] + 1745057227773[1686145264193] + 1430433470707[1921044951457] - 1027813353382[2401808546428] - 993823298739[5345202735279/2] - 1053919165144[3486467472132] + 463817352739[3621667588383] + 1999905303948[5142102426318] + 1368187568227[5602034310634] + 2378564121418[5688475754312] - 783006883663[6202471697443] + 1415985932436[7580898173568] + 2596112296801[15962244871097] - 121109185327[18104444568932] + 2538113946637[18203022356332] + 233665967783[23064526452057] + 838739800829[74963671338441/2] - 562789014113[38057255532937] + 1691756801872[69971515635443] - 2492641396317[180481359172943] + 1540241134881[240926005152903] + 536745742237[258703261539143] + 38727830367[389928169608307] - 517173422907[686308367425978] + 1768427532407[2549868113472987] + 1557551763814[4439568777547493] + 1044205452522[8489727499760268]
Lehmer's measure	2.01530
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 06 August 2006.] Download references as BibTeΧ