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The development of modern genome editing tools has enabled researchers to make such edits with high precision, but has left unsolved the problem of designing these edits. As a solution, we propose Ledidi, a computational method that rephrases the discrete task of designing genomic edits as a continuous optimization problem where the goal is to produce the desired outcome as measured by one or more predictive models using as few edits from an initial template sequence as possible. Ledidi can be paired with almost any trained machine learning model and, when applied across dozens of such models, can quickly design edits to precisely control predicted transcription factor binding, chromatin accessibility, transcription, and enhancer activity across several species. After demonstrating these capabilities, we used Ledidi to design cell type-specific enhancers and validated the designs using STARR-seq. We found that not only did the designs qualitatively induce cell type-specificity, but they also quantitatively controlled regulatory strength, with some designed enhancers exhibiting far greater activity than any naturally occurring enhancer.