Concise high precision approximation for the complete elliptic integral of the first kind

<abstract><p>In this paper, we obtain a concise high-precision approximation for $ \mathcal{K}(r) $:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \frac{2}{\pi }\mathcal{K}(r){\rm{ }}&gt;{\rm{ }}\frac{22\left( r^{\prime }\right) ^{2}+84r^{\prime }+22}{7\left( r^{\prime }\right) ^{3}+57\left( r^{\prime }\right) ^{2}+57r^{\prime }+7}, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>which holds for all $ r\in (0, 1) $, where $ \mathcal{K}(r) $ is complete elliptic integral of the first kind and $ r^{\prime } = \sqrt{1-r^{2}} $.</p></abstract>