 Symmetric rational mixture sampler

The displayed mixture distribution combines a Standard Cauchy distribution and a symmetric law whose magnitude has cumulative distribution function $r^2/(1+r^2)$ for $r\ge 0$. Use independent uniforms $U,V,H$: if $U\le p$, return $\tan (\pi (V-1/2))$; otherwise return a fair independent sign times $\sqrt{V/(1-V)}$. The magnitude density is $2r/(1+r^2{)}^2$ and splitting its mass equally between signs gives $|x|/(1+x^2{)}^2$. An alternative rejection sampling proposal is standard Cauchy, with envelope constant $M=p+\pi (1-p)/2$, because $|x|/(1+x^2)\le 1/2$.