The Hwang equation is not the most efficient Machin like formula to compute $\pi$. The following formula
$${\pi \over 4} = 32 \, \hbox{arctan}({1\over 40}) - \hbox{arctan}\biggl( {38035138859000075702655846657186322249216830232319 \over 2634699316100146880926635665506082395762836079845121}\biggr) $$ has a Lehmer's measure around $1.16751$ thus beating Hwang's formula (with Lehmer's measure $1.51240$).
Abrarov and Quine gave a formula with Lehmer's measure $0.245319$ last summer, together with the relevant algorithms, in their paper An iteration procedure for a two-term Machin-like formula for pi with small Lehmer’s measure. That formula provides 16 new digits of $\pi$ per term increment thus beating the famous Chudnovsky formula.
At $k=27$ the Abrarov-Quine formula gives a fraction that has a numerator with $522\,185\,807$ digits and a denominator with $522\,185\,816$ digits. This type of fractions with huge numbers can also be obtained from the Borwein’s integral as shown in the preprint by Uwe Bäsel and Robert Baillie Sinc integrals and tiny numbers such that a formula for $\pi$ can be written with $453\,130\,145$ and $453\,237\,170$ digits in numerator and denominator, respectively.
