Zernike moments are useful tools in pattern recognition and image analysis. However, direct computation of these moments is very expensive, limiting their use as feature descriptors especially at high orders. The existing methods by employing quantized polar coordinate systems not only save the computational time, but also reduce the accuracy of the moments. In this paper, we propose a hybrid algorithm, which re-organize Zernike moments with any order and repetition as a linear combination of Fourier-Mellin moments, to calculate Zernike moments at high orders fast and accurately. Firstly, arbitrary precision arithmetic is employed to preserve accuracy. Secondly, the property of symmetry is applied to the Fourier-Mellin moments to reduce their computational cost. Thirdly, the recursive relations of Zernike polynomial coefficients are used to speed up their computation. Experimental results reveal that the proposed method is more efficient than the other methods.