 Nearest-integer stopping for amplitude amplification

For known initial success probability $p\in (0,1)$, this choice of iteration count ensures $|(2r+1)\theta -\pi /2|\le \theta$. Hence the success probability after amplitude amplification is at least ${\cos }^2\theta =1-p$. Counting initial state preparation and the forward and inverse preparation in each iterate gives $2r+1$ preparation-unitary applications, asymptotic to $\pi /(2\sqrt{p})$ as $p\to 0$.