Symmetric rational mixture sampler ($f_p(x)=\frac{p}{\pi (1+x^2)}+(1-p)\frac{|x|}{(1+x^2{)}^2}$)

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 $|x|/(1+x^2)\le 1/2$.