M000000312

$$\pi=-2892581443936\arctan\left(\frac{1}{2056424542203}\right)+1036678168064\arctan\left(\frac{2}{4414919775989}\right)-6678116952764\arctan\left(\frac{1}{2292939666693}\right)+9210911996428\arctan\left(\frac{2}{8471235750041}\right)+9426402664388\arctan\left(\frac{1}{5727066010185}\right)+202798758392\arctan\left(\frac{1}{5945958754193}\right)+3884884070724\arctan\left(\frac{1}{7758917519457}\right)+11472711022212\arctan\left(\frac{1}{10546511281730}\right)+1285092665872\arctan\left(\frac{1}{11244667034168}\right)+1655412598400\arctan\left(\frac{2}{23780027088431}\right)+8304920615028\arctan\left(\frac{2}{28456536638489}\right)+6659750656564\arctan\left(\frac{1}{15962244871097}\right)+7717003231748\arctan\left(\frac{1}{24405694099301}\right)+6797581707204\arctan\left(\frac{1}{33308851938318}\right)-362923517408\arctan\left(\frac{1}{41887379804066}\right)-6269495573392\arctan\left(\frac{1}{42859110883727}\right)-1764953810744\arctan\left(\frac{1}{69971515635443}\right)-248110566024\arctan\left(\frac{1}{94465125845354}\right)-1092975334228\arctan\left(\frac{1}{158682175700807}\right)-9369829543136\arctan\left(\frac{1}{187458398552950}\right)+470484427916\arctan\left(\frac{1}{211773236092865}\right)-860486452340\arctan\left(\frac{1}{258213230677943}\right)+650394676108\arctan\left(\frac{1}{370392816588001}\right)+5859341015896\arctan\left(\frac{1}{758341235653128}\right)-4497065740192\arctan\left(\frac{1}{1427056404016693}\right)+14603462597864\arctan\left(\frac{1}{2270451385422943}\right)+4897681109672\arctan\left(\frac{1}{4204987898062723}\right)$$

Compact formula	- 2892581443936[2056424542203] + 1036678168064[4414919775989/2] - 6678116952764[2292939666693] + 9210911996428[8471235750041/2] + 9426402664388[5727066010185] + 202798758392[5945958754193] + 3884884070724[7758917519457] + 11472711022212[10546511281730] + 1285092665872[11244667034168] + 1655412598400[23780027088431/2] + 8304920615028[28456536638489/2] + 6659750656564[15962244871097] + 7717003231748[24405694099301] + 6797581707204[33308851938318] - 362923517408[41887379804066] - 6269495573392[42859110883727] - 1764953810744[69971515635443] - 248110566024[94465125845354] - 1092975334228[158682175700807] - 9369829543136[187458398552950] + 470484427916[211773236092865] - 860486452340[258213230677943] + 650394676108[370392816588001] + 5859341015896[758341235653128] - 4497065740192[1427056404016693] + 14603462597864[2270451385422943] + 4897681109672[4204987898062723]
Lehmer's measure	1.98805
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