?????: computable semantics for differentiable programming with higher-order functions and datatypes

Deep learning is moving towards increasingly sophisticated optimization objectives that employ higher-order functions, such as integration, continuous optimization, and root-finding. Since differentiable programming frameworks PyTorch TensorFlow do not have first-class representations of these developers must reason about the semantics manually translate them to code. We present a language, $\lambda_S$, first deliver for derivatives, Lipschitz but nondifferentiable functions. Together, features enable $\lambda_S$ expose differentiable, functions root-finding with automatically computed derivatives. $\lambda_S$'s computable, meaning values can be arbitrary precision, we implement an embedded language in Haskell. use construct novel libraries representing probability distributions, implicit surfaces, generalized parametric surfaces -- all instances datatypes case studies rely on computing derivatives datatypes. In addition modeling existing algorithms, ray tracer without requiring any user-level differentiation code, demonstrate new Hausdorff distance surfaces.