  We use perturbative Symanzik improvement to create a new staggered-quark
action (HISQ) that has greatly reduced one-loop taste-exchange errors, no
tree-level order a^2 errors, and no tree-level order (am)^4 errors to leading
order in the quark's velocity v/c. We demonstrate with simulations that the
resulting action has taste-exchange interactions that are at least 3--4 times
smaller than the widely used ASQTAD action. We show how to estimate errors due
to taste exchange by comparing ASQTAD and HISQ simulations, and demonstrate
with simulations that such errors are no more than 1% when HISQ is used for
light quarks at lattice spacings of 1/10 fm or less. The suppression of (am)^4
errors also makes HISQ the most accurate discretization currently available for
simulating c quarks. We demonstrate this in a new analysis of the psi-eta_c
mass splitting using the HISQ action on lattices where a m_c=0.43 and 0.66,
with full-QCD gluon configurations (from MILC). We obtain a result of~111(5)
MeV which compares well with experiment. We discuss applications of this
formalism to D physics and present our first high-precision results for D_s
mesons.
