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.

In the Maclaurin series of $\arctan(x)$, the $n$'th term is  $x^{2n-1}/(2n-1)$.  When $|x|$ is small, the important part here is the $x^{2n-1}$.  So to get accuracy $\epsilon$, you need $2n-1 \approx \log (\epsilon)/\log (x)$.  Roughly speaking, 
in a Machin-like formula $\sum_j a_j \arctan(1/b_j)$
the number of terms you need to calculate is inversely proportional to the log of the smallest $b_j$.  
