We calculate the strong isospin breaking and QED corrections to meson masses and the hadronic vacuum polarization in an exploratory study on a $64times24^3$ lattice with an inverse lattice spacing of $a^{-1}=1.78$ GeV and an isospin symmetric pion mass of $m_pi=340$ MeV. We include QED in an electro-quenched setup using two different methods, a stochastic and a perturbative approach. We find that the electromagnetic correction to the leading hadronic contribution to the anomalous magnetic moment of the muon is smaller than $1%$ for the up quark and $0.1%$ for the strange quark, although it should be noted that this is obtained using unphysical light quark masses. In addition to the results themselves, we compare the precision which can be reached for the same computational cost using each method. Such a comparison is also made for the meson electromagnetic mass-splittings.
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