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Group the two terms containing $p$ to obtain the mixture distributionThe first component is a Standard Cauchy distribution. The second is normalized becauseFor this symmetric component, let $R=|X|$. Its density is $2r/(1+r^2{)}^2$, and hence its cumulative distribution function and quantile function areThe independent sign is positive or negative with equal probability. For one sample, use the next three independent uniforms $U,V,H$ in the given sequence and returnThe Cauchy branch follows from its quantile function, while the other branch follows from the magnitude law and fair sign just derived. Independence of the branch selection and component draws therefore gives exactly the displayed mixture distribution. Discard the zero-probability endpoint draw $V=1$ to keep the computation finite, and use a fresh block of three uniforms for the next sample. This is the symmetric rational mixture sampler.

One can alternatively use the root solution's rejection sampling with a Standard Cauchy distribution proposal. The density ratio is $p+\pi (1-p)|x|/(1+x^2)$, bounded by $M=p+\pi (1-p)/2$ because $2|x|\le 1+x^2$. Accept the Cauchy draw $Y$ with probability $[p+\pi (1-p)|Y|/(1+Y^2)]/M$ using a fresh independent uniform.