During the design of embedded systems, many design decisions have to be made to trade off between conflicting objectives such as cost, performance, and power. Approximate computing allows to optimize each objective, yet for the sake of accuracy. This means that a functional flaw is allowed to produce an error as long as this is small enough to maintain a feasible operation of the system or guarantee a certain accuracy of the results. In this paper, we propose a new technique for approximate addition optimized for LUT-Based FPGAs with segmented carry chains. Our optimized adder structure is able to a) best exploit artifacts of LUT-Based FPGAs such as unused inputs and b) provide a smaller average error than previously proposed approximate adder structures, as well as c) a reduced critical path delay than dedicated accurate logic in modern FPGAs. We present a novel stochastic error calculus that is able to take into account also non-uniform input distributions and present a detailed comparison of approximate adder structures proposed in literature with our novel LUT-Based approximate arithmetic structure.