Title: Efficient computation of pi by the Newton - Raphson iteration and a two-term Machin-like formula,
Abstract: 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
${1/\left|\beta_2\right|}$ may be significantly smaller than ${1/\beta_1}$, 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.