The configuration model is a cornerstone of statistical assessment of network structure. While the Chung-Lu model is among the most widely used configuration models, it systematically oversamples edges between large-degree nodes, leading to inaccurate statistical conclusions. Although the maximum entropy principle offers unbiased configuration models, its high computational cost has hindered widespread adoption, making the Chung-Lu model an inaccurate yet persistently practical choice. Here, we propose fast and efficient sampling algorithms for the max-entropy-based models by adapting the Miller-Hagberg algorithm. Evaluation on 103 empirical networks demonstrates 10-1000 times speedup, making theoretically rigorous configuration models practical and contributing to a more accurate understanding of network structure.