The Fast Fourier Transformation (FFT), as a high-efficiency algorithm of the Discrete Fourier Transform (DFT), is widely used in Digital Signal Processing (DSP), wireless communication systems, spectrum analysis, and image processing. Approximate computing has shown effectiveness and feasibility to enhance the hardware efficiency of these applications. However, most approximate units in previous works are designed case by case, which has low efficiency and is difficult to find the optimal design. In this paper, a top-down design strategy for approximate floating-point (FP) FFT is proposed, which includes a mantissa bit-width adjustment algorithm and a step-by-step multiplier approximation algorithm. With the mantissa bit-width adjustment algorithm, the approximate 64 FP FFT achieved 50% area reduction and 70% power-delay product (PDP) reduction compared to the exact design with a 60dB Signal Noise Ratio (SNR) requirement, which is also at least 52% and 33% better than the previous approximate FP FFT. After using the step-by-step multiplier approximation algorithm, the approximate mantissa multiplier with an 8-bit fractional part reduced the area and PDP by 81.15% and 93.70%, respectively. The feasibility of the proposed approximate FFT design is verified in the channel estimation module of a wireless communication system, spectrum analysis, and image processing system.