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Linear light processing functions (e.g., routing, splitting, filtering) are key functions requiring configuration to implement on a programmable photonic integrated circuit (PPIC). In recirculating waveguide meshes (which include loop-backs), this is usually done manually. Some previous results describe explorations to perform this task automatically, but their efficiency or applicability is still limited. In this paper, we propose an efficient method that can automatically realize configurations for many light processing functions on a square-mesh PPIC. At its heart is an automatic differentiation subroutine built upon analytical expressions of scattering matrices, that enables gradient descent optimization for functional circuit synthesis. Similar to the state-of-the-art synthesis techniques, our method can realize configurations for a wide range of light processing functions, and multiple functions on the same PPIC simultaneously. However, we do not need to separate the functions spatially into different subdomains of the mesh, and the resulting optimum can have multiple functions using the same part of the mesh. Furthermore, compared to non-gradient or numerical differentiation based methods, our proposed approach achieves 3x time reduction in computational cost.