Is there an efficient circuit implementing the unitary $U|xrangle|0rangle=|xrangleBig(sqrt{1 - x/2^n},|0rangle+sqrt{x/2^n}|1rangleBig)?$

Given an $n$-qubit register $|xrangle$, does there exist an efficient circuit implementing unitary operation $U$ such that

$$U |xrangle|0rangle = |xrangleBig(sqrt{1 – x/2^n}, |0rangle + sqrt{x/2^n}, |1rangleBig)?$$

I’ve found this related question from which the answer suggests to rotate and apply an $arccos$ approximation (which is very complicated, and only provides an approximation). Is there not an exact circuit implementing this from simple gates plus $R_k$?

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The context of this question is trying to implement Algorithm 1 from Quantum speedup of Monte Carlo methods by Ashley Montanaro. They say (paraphrased):

Also observe that $U$ can be implemented efficiently, as it is a controlled rotation of one qubit dependent on the value of $x$ [59]

I did not find the linked reference (Quantum algorithm for approximating partition functions by Wocjan et al) particularly enlightening. And I don’t believe they used the $arccos$ approximation either, as they did not include this in the error analysis. So I am confused as to how $U$ is actually implemented.