Application of a new iterative formula for computing $π$ and nested radicals with roots of $2$
Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine
TL;DR
The paper introduces a novel iterative formula (NIF) to compute $\pi$ and nested radicals built from roots of $2$, recasting the two-term Machin-like π representation into a real-valued recurrence $v_k=\tfrac{1}{2}\left(v_{k-1}-1/v_{k-1}\right)$ that yields $\frac{\pi}{4}=2^{k-1}\arctan\left(\frac{1}{v_1}\right)+\arctan\left(\frac{v_k-1}{v_k+1}\right)$. It connects $\gamma_k$, $\theta_{1,k}$, and $v_k$ to obtain reciprocal-integer forms suitable for high-precision arithmetic and rational approximations, and it presents complementary strategies including rational/2-term approximations, cubic-convergence iteration, nested radicals, and arbitrary-precision schemes with accompanying Mathematica code. The results demonstrate substantial speedups in computing constants like $\theta_{1,k}$ (e.g., up to ~5× faster) and the rapid growth of digits of $\pi$ under the new framework, supporting efficient, scalable calculation of many digits. Overall, the work provides a practical toolkit for high-precision $\pi$ computation and nested-radical evaluations, with clear pathways for implementation and extension in numerical practice.
Abstract
In this work, we obtain an iterative formula that can be used for computing digits of $π$ and nested radicals of kind $c_n/\sqrt{2 - c_{n - 1}}$, 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 $π$ can be generated and approximated. Some examples with Mathematica codes are presented.