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A somewhat surprising request on X validated about the inverse cdf representation of a warped Cauchy distribution. I had not come across this distribution, but its density being

$f_{WC}(\theta;\gamma)=\sum_{n=-\infty}^\infty \frac{\gamma}{\pi(\gamma^2+(\theta+2\pi n)^2)}\mathbb I_{ -\pi<\theta<\pi}$

means that it is the superposition of shifted Cauchys on the unit circle (with nice complex representations). As such, it is easily simulated by re-shifting a Cauchy back to (-π,π), i.e. using the inverse transform

$\theta = [\gamma\tan(\pi U-\pi/2)+\pi]\ \text{mod}\,(2\pi) - \pi$