  We present a calculation of the $D\to K \ell \nu$ and $D\to\pi \ell \nu$
semileptonic form factors at $q^2=0$, which enable determinations of the CKM
matrix elements $\lvert{V_{cs}}\rvert$ and $\lvert{V_{cd}}\rvert$,
respectively. We use gauge-field configurations generated by the MILC
collaboration with four flavors of highly-improved staggered (HISQ) quarks,
analyzing several ensembles including those with physical pion masses and
approximate lattice spacings ranging from 0.12~fm to 0.042~fm. We also use the
HISQ action for the valence quarks. We employ twisted boundary conditions to
calculate the form factors at zero momentum transfer directly. We use
heavy-light-meson chiral perturbation theory modified for energetic pions and
kaons, and supplemented by terms to describe the lattice-spacing dependence, to
obtain preliminary results at the physical point and in the continuum limit.
