  The conformal mapping $f(z)=(z+1)^2 $ from $\mathbb{D}$ onto the standard
cardioid has a homeomorphic extension of finite distortion to entire
$\mathbb{R}^2 .$ We study the optimal regularity of such extensions, in terms
of the integrability degree of the distortion and of the derivatives, and these
for the inverse. We generalize all outcomes to the case of conformal mappings
from $\mathbb{D}$ onto cardioid-type domains.
