Accurate estimation of discharge capacity in flap gates plays a fundamental role in flow measurement and hydraulic control within irrigation networks. In this research, three advanced data-driven models were developed for predicting the dimensionless discharge capacity in rectangular flap gates under free flow conditions. To this end, a comprehensive laboratory dataset incorporating effective dimensionless hydraulic and geometric parameters was utilized to construct models based on Multivariate Adaptive Regression Splines (MARS), eXtreme Gradient Boosting (XGBoost), and Gene Expression Programming (GEP). The results indicated that the MARS model achieved the highest prediction accuracy, with root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²) values of 0.00578, 0.00818, and 0.9978, respectively, demonstrating its superior capability in modeling complex nonlinear interactions in this type of hydraulic measurement problem. The XGBoost model, owing to its tree-based structure founded on gradient boosting, also exhibited reliable performance with performance metrics of RMSE = 0.02153, MAE = 0.01495, and R² = 0.9877. In contrast, the GEP model, despite providing explicit analytical relationships, showed relatively lower accuracy. The objective function criterion (OBF) further confirmed the superiority of the MARS and XGBoost models over traditional nonlinear regression equations and previously reported machine learning models. Sensitivity analysis and Shapley Additive Explanations (SHAP) revealed that the relative upstream flow head was the most influential variable in controlling the dimensionless discharge, which is consistent with fundamental hydraulic theory. Additionally, the gamma test played an effective role in selecting the optimal combination of input variables and improving model smoothness. Model uncertainty was evaluated using confidence limits of prediction errors. The results indicate that the XGBoost model yields the most stable and reliable predictions, with the smallest variability in error bounds, making it well suited for accurate and generalizable estimation of flap gate discharge capacity.