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In our recent publication we have proposed a new methodology fordetermination of the two-term Machin-like formula for pi with small argumentsof 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 arational number, dependent upon ${\beta_1}$ and $k$. Although${1/\left|\beta_2\right|}$ may be significantly smaller than ${1/\beta_1}$, thelarge numbers in the numerator and denominator of $\beta_2$ decelerate thecomputation. In this work we show how this problem can be effectively resolvedby the Newton--Raphson iteration method.

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