Active deployment of machine learning systems sets a task of their protection against different types of attacks that threaten confidentiality, integrity and accessibility of both processed data and trained models. One of the promising ways for such protection is the development of privacy-preserving machine learning systems, that use homomorphic encryption schemes to protect data and models. However, such schemes can only process polynomial functions, which means that we need to construct polynomial approximations for nonlinear functions used in neural models. The goal of this paper is the construction of precise approximations of several widely used neural network activation functions while limiting the degree of approximation polynomials as well as the evaluation of the impact of the approximation precision on the resulting value of the whole neural network. In contrast to the previous publications, in the current paper we study and compare different ways for polynomial approximation construction, introduce precision metrics, present exact formulas for approximation polynomials as well as exact values of corresponding precisions. We compare our results with the previously published ones. Finally, for a simple convolutional network we experimentally evaluate the impact of the approximation precision on the bias of the output neuron values of the network from the original ones. Our results show that the best approximation for ReLU could be obtained with the numeric method, and for the sigmoid and hyperbolic tangent – with Chebyshev polynomials. At the same time, the best approximation among the three functions could be obtained for ReLU. The results could be used for the construction of polynomial approximations of activation functions in privacy-preserving machine learning systems.