LLVM code optimisation for automatic differentiation: when forward and reverse mode lead in the same direction

Both forward and reverse mode automatic differentiation derive a model function as used for gradient descent automatically. Reverse mode calculates all derivatives in one run, whereas forward mode requires rerunning the algorithm with respect to every variable for which the derivative is needed. To allow for in-database machine learning, we have integrated automatic differentiation as an SQL operator inside the Umbra database system. To benchmark code-generation to GPU, we implement forward as well as reverse mode automatic differentiation. The inspection of the optimised LLVM code shows that nearly the same machine code is executed after the generated LLVM code has been optimised. Thus, both modes yield similar runtimes but different compilation times.