Accurate nonlinear computation is a key challenge in privacy-preserving machine learning (PPML). Most existing frameworks approximate it through linear operations, resulting in significant precision loss. This paper proposes an efficient, verifiable and accurate security 2-party logistic regression framework (EVA-S2PLoR), which achieves accurate nonlinear function computation through a novel secure element-wise multiplication protocol and its derived protocols. Our framework primarily includes secure 2-party vector element-wise multiplication, addition to multiplication, reciprocal, and sigmoid function based on data disguising technology, where high efficiency and accuracy are guaranteed by the simple computation flow based on the real number domain and the few number of fixed communication rounds. We provide secure and robust anomaly detection through dimension transformation and Monte Carlo methods. EVA-S2PLoR outperforms many advanced frameworks in terms of precision (improving the performance of the sigmoid function by about 10 orders of magnitude compared to most frameworks) and delivers the best overall performance in secure logistic regression experiments.