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In this era of noisy intermediate-scale quantum (NISQ) computing, systematic miscalibrations, drift, and crosstalk in the control of quantum bits can lead to a coherent form of error which has no classical analog. Such errors severely limit the performance of quantum algorithms in an unpredictable manner, and mitigating their impact is necessary for realizing reliable quantum computations. Randomized compiling is a protocol designed to overcome these performance limitations by converting coherent errors into stochastic noise, dramatically reducing unpredictable errors in quantum algorithms and enabling accurate predictions of algorithmic performance from error rates measured via cycle benchmarking. We demonstrate significant performance gains under randomized compiling for the algorithms including the four-qubit quantum Fourier transform and for random circuits of variable depth on a superconducting quantum processor. Additionally, we accurately predict algorithm performance using experimentally-measured error rates. Our results demonstrate that randomized compiling can be utilized to maximally-leverage and predict the capabilities of modern-day noisy quantum processors, paving the way forward to scalable quantum computing.