TITLE: Why does it help to use $640320^3 = 8\cdot 100100025\cdot 327843840$ when you calculate $\pi$?
QUESTION [5 upvotes]: A Ramanujan-type formula due to the Chudnovsky brothers used to break a world record for computing the most digits of $\pi$:
$$ \frac{1}{\pi} = \frac{1}{53360 \sqrt{640320}} \sum_{n=0}^\infty (-1)^n \frac{(6n)!}{n!^3(3n)!} \times \frac{13591409 + 545140134n}{640320^{3n}} \tag{$*$}
$$
For implementations, it may help to use $640320^3 = 8\cdot 100100025\cdot 327843840$.

How does this help?

REPLY [2 votes]: $$e^{\pi\sqrt{163}}\ \simeq\ 640,320^3+744\qquad\iff\qquad\pi\ \simeq\ \frac{\ln(640,320^3+744)}{\sqrt{163}}$$ See Heegner number for more details. The precision is $30$ decimals.