Training large-scale neural networks is heavily constrained by GPU memory. In order to circumvent this limitation, gradient checkpointing, or recomputation is a powerful technique. There is active research in this area with methods such as Checkmake or Moccasin. However, both Checkmate and Moccasin rely on mixed integer linear programming or constraint programming, resulting in limited scalability due to their exponentially large search space. This paper proposes a novel algorithm for recomputation (FastSA) based on a simulated annealing heuristic that achieves comparable or even better solutions than state-of-the-art alternatives. FastSA can optimize computational graphs with thousands of nodes within 3 to 30 seconds, several orders of magnitude faster than current solutions. We applied FastSA to PyTorch models and verified its effectiveness through popular large vision and text models, including recent language models with the transformer architecture. The results demonstrate significant memory reductions by 73% with extra 18% computational overheads on average. Our experiments demonstrate the practicality and effectiveness of our recomputation algorithm, further highlighting its potential for wide application in various deep learning domains.