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We use perturbative Symanzik improvement to create a new staggered-quarkaction (HISQ) that has greatly reduced one-loop taste-exchange errors, notree-level order a^2 errors, and no tree-level order (am)^4 errors to leadingorder in the quark's velocity v/c. We demonstrate with simulations that theresulting action has taste-exchange interactions that are at least 3--4 timessmaller than the widely used ASQTAD action. We show how to estimate errors dueto taste exchange by comparing ASQTAD and HISQ simulations, and demonstratewith simulations that such errors are no more than 1% when HISQ is used forlight quarks at lattice spacings of 1/10 fm or less. The suppression of (am)^4errors also makes HISQ the most accurate discretization currently available forsimulating c quarks. We demonstrate this in a new analysis of the psi-eta_cmass 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 thisformalism to D physics and present our first high-precision results for D_smesons.Comment: 21 pages, 8 figures, 5 table

Highly improved staggered quarks on the lattice with applications to charm physics