  We study the impact of hypercubic (HYP) smearing on the size of taste
breaking for staggered fermions, comparing to unimproved and to asqtad-improved
staggered fermions. As in previous studies, we find a substantial reduction in
taste-breaking compared to unimproved staggered fermions (by a factor of 4-7 on
lattices with spacing $a\approx 0.1 $fm). In addition, we observe that
discretization effects of next-to-leading order in the chiral expansion (${\cal
O}(a^2 p^2)$) are markedly reduced by HYP smearing. Compared to asqtad valence
fermions, we find that taste-breaking in the pion spectrum is reduced by a
factor of 2.5-3, down to a level comparable to the expected size of generic
${\cal O}(a^2)$ effects. Our results suggest that, once one reaches a lattice
spacing of $a\approx 0.09 $fm, taste-breaking will be small enough after HYP
smearing that one can use a modified power counting in which ${\cal O}(a^2) \ll
{\cal O}(p^2)$, simplify fitting to phenomenologically interesting quantities.
