How can we determine the accuracy of an equation (or algorithm) to generate an approximations of $\pi$?

For exampl:

Ramanujan: $$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. $$

Chudnovsky brothers: $$ \frac{1}{\pi} = \frac{1}{53360 \sqrt{640320}} \sum_{n=0}^\infty (-1)^n \frac{(6n)!}{n!^3(3n)!} \times \frac{13591409 + 545140134n}{640320^{3n}}$$ or Bailey–Borwein–Plouffe formula

$$ \pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right]$$

If a new equation arises to calculate $\pi$, How can we determine the accuracy of this?


According to this answer on a stack overflow question of the same topic:

Historically, the standard approach for verifying that computed digits are correct is to recompute the digits using a second algorithm. So if either computation goes bad, the digits at the end won't match.

This answer is regarding the qualifications for world record calculations of $\pi$. This method allows you to confirm that two methods produce the same result, which is usually a good representation of whether or not that result is correct.

According to the same answer,

Nowadays, things are a bit different. In the last three world records, instead of performing two computations, we performed only one computation using the fastest known formula (Chudnovsky Formula):

$$\frac{1}{\pi}=\frac{\sqrt{1005}}{4270934400} \sum_{k=0}^{\infty} (-1)^k \frac{(6k)!}{(k!)^3(3k)!}\frac{13591409+545140134k}{640320^k} $$

Then we verify the binary digits using the BBP formulas for digit extraction.

$$\pi= \sum_{i=0}^\infty \left[ \frac{1}{16^i}\left( \frac{4}{8i+1}-\frac{2}{8i+4}-\frac{1}{8i+5}-\frac{1}{8i+6} \right) \right]$$

This formula allows you to compute arbitrary binary digits without computing all the digits before it. So it is used to verify the last few computed binary digits.


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