Efficient computation of non-polynomial functions in homomorphic encryption schemes is important for secure operations across various fields. Recent studies have focused on developing efficient algorithms for these computations. In this paper, we extend the Remez algorithm to derive an approximate polynomial with minimal relative error, enabling the homomorphic evaluation of non-polynomial functions such as the reciprocal m-th root function \(f(x) = x^{-1/m}\) where \(m\in \mathbb {Z}^+\) and the exponential function \(f(x)=e^x\) . Additionally, we introduce a novel approximate polynomial \(u(x)=\sum _{i=0}^{n}c_i r^{i}x^{mi+1}\) designed to further refine the approximation range for \(f(x)=x^{-1/m}\) . Utilizing these methods, we introduce new algorithms for the homomorphic evaluation of inverse, square root, and exponential functions, which are subsequently implemented with the SEAL library. The experimental results demonstrate the superiority of our algorithms, requiring only \(20\%\) of the execution time for the inverse function and \(15\%\) for the square root function compared to previous method. Furthermore, based on our approximate inverse and exponential algorithms, we propose a new algorithm for the softmax function that exhibits reduced computation time and significantly smaller maximum and average errors.