The Chudnovsky algorithm based on hypergeometric series seems to appear prominently. Why are approaches based around $\arctan$ slower?
Is there some intuitive, conceptual description of the "redundancy" or inefficiency in $\arctan$ based calculations that explain why they are slower?
Sure, the coefficients in the power series of $\arctan$ are $\sim\frac1n$, hence decrease very slowly. So to compute $n$ digits of $\arctan x$ with $x\approx 10^{-k}$, you need $\frac nk$ summands.