Computational efficiency of Machin-like formulae From what I have read, it appears that the most efficient methods of calculating $ \pi $ are Machin-like formulae. And it is known that certain formulas are more efficient than others.
Are there any theoretical properties (rates of convergence, etc) of these formulas that make them more efficient than others?
Like, why is
$$
\frac{\pi}{4} = 183\arctan\frac{1}{239} + 32\arctan\frac{1}{1023} - 68\arctan\frac{1}{5832} + 12\arctan\frac{1}{110443} - 12\arctan\frac{1}{4841182} - 100\arctan\frac{1}{6826318}
$$
more efficient than
$$ \frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3} $$
(as claimed by Wikipedia)
Or is this a purely empirical result?
I presume that the calculation of arctan is done using the Maclaurin series.

With the Maclaurin series, after the first few summations, each successive sum will be slightly less than $1/4$ the value of the previous one when calculating $\arctan(1/2)$.
However, with $\arctan(1/239)$, each successive term in the series is less than $1/57,000$ the size as the previous term, so you are able to generate multiple additional decimal places with each additional term in the sequence.
However, to calculate millions of digits very quickly (using a computer, of course), it is better to use other algorithms.
See Chudnovsky algorithm (according to wikipedia, this was used to calculate 10 trillion digits of pi)
