Memristors are non-volatile memory devices that are also capable of colocation computations. General-purpose computations can use memristors to approximate arbitrary functions with neural networks or can use memristors to model basic gate circuits that then perform arbitrary Boolean logic calculations. However, the use of memristors to approximate arbitrary functions does not have controllable errors and the use of memristors to model basic gate circuits is slower than conventional digital circuits. This paper presents a general-purpose approximate computing paradigm for memristors and a memristor based hardware architecture, general-purpose field programmable synapse array (GP-FPSA), that combines the advantages of these two methods for efficient general-purpose approximate computing with controllable errors. A universal approximating construction method is used to resolve the large, uncontrollable error of directly training a neural network for approximations. Then, the model control flow splits complicated functions to reduce the construction cost. The memristor-based architecture significantly improves the computational power for general-purpose computing.