Application of a new iterative formula for computing $\pi$ and nested radicals with roots of $2$

In this work, we obtain an iterative formula that can be used for computing digits of $\pi$ and nested radicals of kind $c_n\mathrm{/}\sqrt{2-c_{n-1}}$, where $c_0=0$ and $c_n=\sqrt{2+c_{n-1}}$. We also show how with the help of this iterative formula, the two-term Machin-like formulas for $\pi$ can be generated and approximated. Some examples with Mathematica codes are presented.