The implementation of accurate nonlinear operators (e.g., sigmoid function) on heterogeneous datasets is a key challenge in privacy-preserving machine learning (PPML). Most existing frameworks approximate it through linear operations, which not only result in significant precision loss but also introduce substantial computational overhead. This paper proposes an efficient, verifiable, and accurate security 2-party logistic regression framework (EVA-S2PLoR), which achieves accurate nonlinear function computation through a subtly secure hadamard product protocol and its derived protocols. All protocols are based on a practical semi-honest security model, which is designed for decentralized privacy-preserving application scenarios that balance efficiency, precision, and security. High efficiency and precision are guaranteed by the asynchronous computation flow on floating point numbers and the few number of fixed communication rounds in the hadamard product protocol, where robust anomaly detection is promised by 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. Moreover, EVA-S2PLoR delivers the best overall performance in secure logistic regression experiments with training time reduced by over 47.6% under WAN settings and a classification accuracy difference of only about 0.5% compared to the plaintext model.