Recent efforts for improving the performance of neural network (NN) accelerators that meet today's application requirements have given rise to a new trend of logic-based NN inference relying on fixed function combinational logic. Mapping such large Boolean functions with many input variables and product terms to digital signal processors (DSPs) on Field-programmable gate arrays (FPGAs) needs a novel framework considering the structure and the reconfigurability of DSP blocks during this process. The proposed methodology in this paper maps the fixed function combinational logic blocks to a set of Boolean functions where Boolean operations corresponding to each function are mapped to DSP devices rather than look-up tables (LUTs) on the FPGAs to take advantage of the high performance, low latency, and parallelism of DSP blocks. % This paper also presents an innovative design and optimization methodology for compilation and mapping of NNs, utilizing fixed function combinational logic to DSPs on FPGAs employing high-level synthesis flow. % Our experimental evaluations across several \REVone{datasets} and selected NNs demonstrate the comparable performance of our framework in terms of the inference latency and output accuracy compared to prior art FPGA-based NN accelerators employing DSPs.