M000000189

$$\pi=-10490168\arctan\left(\frac{1}{36226953}\right)+44691688\arctan\left(\frac{1}{78739287}\right)+259478548\arctan\left(\frac{2}{204437211}\right)-156327472\arctan\left(\frac{2}{322564791}\right)-41130520\arctan\left(\frac{2}{354124475}\right)+411917852\arctan\left(\frac{1}{193788912}\right)+5375732\arctan\left(\frac{1}{201229582}\right)-165798056\arctan\left(\frac{1}{284862638}\right)-55392476\arctan\left(\frac{1}{1085592682}\right)+15899536\arctan\left(\frac{1}{5667666318}\right)-8154604\arctan\left(\frac{1}{7427844818}\right)+137362488\arctan\left(\frac{2}{75525552427}\right)+110050500\arctan\left(\frac{1}{57821838732}\right)-1637580\arctan\left(\frac{1}{94111829193}\right)+98638048\arctan\left(\frac{2}{1087840667761}\right)+174589052\arctan\left(\frac{1}{169838669284032}\right)$$

Compact formula	`- 10490168[36226953] + 44691688[78739287] + 259478548[204437211/2] - 156327472[322564791/2] - 41130520[354124475/2] + 411917852[193788912] + 5375732[201229582] - 165798056[284862638] - 55392476[1085592682] + 15899536[5667666318] - 8154604[7427844818] + 137362488[75525552427/2] + 110050500[57821838732] - 1637580[94111829193] + 98638048[1087840667761/2] + 174589052[169838669284032]`
Lehmer's measure	1.73482
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 12 August 2006.] Download references as BibTeΧ