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Flavor SU(3) symmetry, including $30\%$ first order SU(3) breaking, has beenshown to describe adequately a vast amount of data for charmed meson decays totwo pseudoscalar mesons and to a vector and a pseudoscalar meson. We review arecent dramatic progress achieved by applying a high order perturbationexpansion in flavor SU(3) breaking and treating carefully isospin breaking. Weidentify a class of U-spin related $D^0$ decays to pairs involving chargedpseudoscalar or vector mesons, for which high-precision nonlinear amplituderelations are predicted. Symmetry breaking terms affecting these relations arefourth order U-spin breaking, and terms which are first order in isospinbreaking and second order in U-spin breaking. The predicted relations are shownto hold experimentally at a precision varying between $10^{-3}$ and $10^{-4}$,in agreement with estimates of high order terms. We also discuss amplituderelations for $D^0$ decays to pairs of neutral pseudoscalar mesons, andrelations for rate asymmetries between decays involving $K^0_S$ and $K^0_L$which hold up to second order U-spin breaking.

SU(3) in D decays: From $30\%$ symmetry breaking to $10^{-4}$ precision