  Loewner driving functions encode simple curves in 2-dimensional simply
connected domains by real-valued functions. We prove that the Loewner driving
function of a $C^{1,\beta}$ curve (differentiable parametrization with
$\beta$-H\"older continuous derivative) is in the class $C^{1,\beta-1/2}$ if
$1/2<\beta\leq 1$, and in the class $C^{0,\beta + 1/2}$ if $0 \leq \beta \leq
1/2$. This is the converse of a result of Carto Wong and is optimal. We also
introduce the Loewner energy of a rooted planar loop and use our regularity
result to show the independence of this energy from the basepoint.
