Spiking neural networks (SNNs) have been widely used due to their strong biological interpretability and high-energy efficiency. With the introduction of the backpropagation algorithm and surrogate gradient, the structure of SNNs has become more complex, and the performance gap with artificial neural networks (ANNs) has gradually decreased. However, most SNN hardware implementations for field-programmable gate arrays (FPGAs) cannot meet arithmetic or memory efficiency requirements, which significantly restricts the development of SNNs. They do not delve into the arithmetic operations between the binary spikes and synaptic weights or assume unlimited on-chip RAM resources using overly expensive devices on small tasks. To improve arithmetic efficiency, we analyze the neural dynamics of spiking neurons, generalize the SNN arithmetic operation to the multiplex-accumulate operation, and propose a high-performance implementation of such operation by utilizing the DSP48E2 hard block in Xilinx Ultrascale FPGAs. To improve memory efficiency, we design a memory system to enable efficient synaptic weights and membrane voltage memory access with reasonable on-chip RAM consumption. Combining the above two improvements, we propose an FPGA accelerator that can process spikes generated by the firing neurons on-the-fly (FireFly). FireFly is the first SNN accelerator that incorporates DSP optimization techniques into SNN synaptic operations. FireFly is implemented on several FPGA edge devices with limited resources but still guarantees a peak performance of 5.53 TOP/s at 300 MHz. As a lightweight accelerator, FireFly achieves the highest computational density efficiency compared with existing research using large FPGA devices.