An iteration procedure for a twoterm Machinlike formula for pi with small Lehmer's measure
Abstract
In this paper we present a twoterm Machinlike formula for pi \[\frac{\pi}{4} = 2^{k  1}\arctan\left(\frac{1}{u_1}\right) + \arctan\left(\frac{1}{u_2}\right)\] with small Lehmer's measure $e \approx 0.245319$ and describe iteration procedure for simplified determination of the required rational number $u_2$ at $k = 27$ and $u_1 = 85445659$. With these results we obtained a formula that has no irrational numbers involved in computation and provides $16$ digits of pi at each increment by one of the summation terms. This is the smallest Lehmer's measure ever reported for the Machinlike formulas for pi.
 Publication:

arXiv eprints
 Pub Date:
 June 2017
 arXiv:
 arXiv:1706.08835
 Bibcode:
 2017arXiv170608835A
 Keywords:

 Mathematics  General Mathematics;
 11Y60
 EPrint:
 20 pages