Uniforms consumed by the polar Gaussian method

A normal random number generator uses the polar (Marsaglia) rejection method: draw two uniforms on $(-1,1)$, reject unless they fall inside the unit disc, then transform. On average, how many uniform draws are used per Gaussian produced?

$4/\pi\approx1.27$ if both Gaussians from each accepted pair are used, or $8/\pi\approx2.55$ if one is discarded. Each attempt uses $2$ uniforms and is accepted with probability equal to the disc's share of the square, $\pi/4$, so a pair costs $8/\pi$ uniforms on average and yields two independent normals.