Efficient computation of pi by the Newton - Raphson iteration and a two-term Machin-like formula

In our recent publication we have proposed a new methodology for determination of the two-term Machin-like formula for pi with small arguments of the arctangent function of kind $$\frac{\pi }{4}=2^{k-1}\arctan \left(\frac{1}{{\beta }_1}\right)+\arctan \left(\frac{1}{{\beta }_2}\right),$$ where $k$ and ${\beta }_1$ are some integers and ${\beta }_2$ is a rational number, dependent upon ${\beta }_1$ and $k$. Although $1\mathrm{/}\mid {\beta }_2\mid$ may be significantly smaller than $1\mathrm{/}{\beta }_1$, the large numbers in the numerator and denominator of ${\beta }_2$ decelerate the computation. In this work we show how this problem can be effectively resolved by the Newton–Raphson iteration method.