We present a new method, called $x$-cut cosmic shear, which optimally removes sensitivity to poorly modeled scales from the two-point cosmic shear signal. We show that the $x$-cut cosmic shear covariance matrix can be computed from the correlation function covariance matrix in a few minutes, enabling a likelihood analysis at virtually no additional computational cost. Further we show how to generalize $x$-cut cosmic shear to galaxy-galaxy lensing. Performing an $x$-cut cosmic shear analysis of the Dark Energy Survey Year 1 (DESY1) shear data, we reduce the error on $S_8= sigma_8 (Omega_m / 0.3) ^ {0.5}$ by $32 %$ relative to a correlation function analysis with the same priors and angular scale cut criterion, while showing our constraints are robust to different baryonic feedback models. Largely driven by information at small angular scales, our result, $S_8= 0.734 pm 0.026$, yields a $2.6 sigma$ tension with the Planck Legacy analysis of the cosmic microwave background. As well as alleviating baryonic modelling uncertainties, our method can be used to optimally constrain a large number of theories of modified gravity where computational limitations make it infeasible to model the power spectrum down to extremely small scales. The key parts of our code are made publicly available.
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