As atmospheric neutrinos propagate through the Earth, vacuumlike oscillations are modified by Standard Model neutral- and charged-current interactions with electrons. Theories beyond the Standard Model introduce heavy, TeV-scale bosons that can produce nonstandard neutrino interactions. These additional interactions may modify the Standard Model matter effect producing a measurable deviation from the prediction for atmospheric neutrino oscillations. The result described in this paper constrains nonstandard interaction parameters, building upon a previous analysis of atmospheric muon-neutrino disappearance with three years of IceCube DeepCore data. The best fit for the muon to tau flavor changing term is εμτ=−0.0005, with a 90% C.L. allowed range of −0.0067<εμτ<0.0081. This result is more restrictive than recent limits from other experiments for εμτ. Furthermore, our result is complementary to a recent constraint on εμτ using another publicly available IceCube high-energy event selection. Together, they constitute the world's best limits on nonstandard interactions in the μ−τ sector.