M000000320

$$\pi=645457936156\arctan\left(\frac{1}{1259851011582}\right)+1208311485536\arctan\left(\frac{1}{1686145264193}\right)+1888889935624\arctan\left(\frac{1}{2267426627682}\right)-520426090140\arctan\left(\frac{1}{2401808546428}\right)-2057528623548\arctan\left(\frac{2}{5345202735279}\right)+473603512404\arctan\left(\frac{1}{3621667588383}\right)+3053824469092\arctan\left(\frac{1}{5142102426318}\right)+851014145320\arctan\left(\frac{1}{5602034310634}\right)+3412910967232\arctan\left(\frac{1}{5688475754312}\right)-1280607987240\arctan\left(\frac{1}{6202471697443}\right)+1455130571096\arctan\left(\frac{1}{7580898173568}\right)+2454994156856\arctan\left(\frac{1}{12575227775843}\right)+3640245302280\arctan\left(\frac{1}{15962244871097}\right)-130895344992\arctan\left(\frac{1}{18104444568932}\right)+3055287369544\arctan\left(\frac{1}{18203022356332}\right)+243452127448\arctan\left(\frac{1}{23064526452057}\right)+868098279824\arctan\left(\frac{2}{74963671338441}\right)-1079962437020\arctan\left(\frac{1}{38057255532937}\right)+2218716384444\arctan\left(\frac{1}{69971515635443}\right)-487814943912\arctan\left(\frac{1}{79507238248932}\right)-3536774401796\arctan\left(\frac{1}{180481359172943}\right)+2096559196448\arctan\left(\frac{1}{240926005152903}\right)-478445592540\arctan\left(\frac{1}{389928169608307}\right)+2305173274644\arctan\left(\frac{1}{2549868113472987}\right)+1537979444484\arctan\left(\frac{1}{4439568777547493}\right)+1590737354424\arctan\left(\frac{1}{8489727499760268}\right)+39144638660\arctan\left(\frac{1}{74295753693510382}\right)$$

Compact formula	645457936156[1259851011582] + 1208311485536[1686145264193] + 1888889935624[2267426627682] - 520426090140[2401808546428] - 2057528623548[5345202735279/2] + 473603512404[3621667588383] + 3053824469092[5142102426318] + 851014145320[5602034310634] + 3412910967232[5688475754312] - 1280607987240[6202471697443] + 1455130571096[7580898173568] + 2454994156856[12575227775843] + 3640245302280[15962244871097] - 130895344992[18104444568932] + 3055287369544[18203022356332] + 243452127448[23064526452057] + 868098279824[74963671338441/2] - 1079962437020[38057255532937] + 2218716384444[69971515635443] - 487814943912[79507238248932] - 3536774401796[180481359172943] + 2096559196448[240926005152903] - 478445592540[389928169608307] + 2305173274644[2549868113472987] + 1537979444484[4439568777547493] + 1590737354424[8489727499760268] + 39144638660[74295753693510382]
Lehmer's measure	2.00586
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