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Recent advanced studies have spent considerable human efforts on optimizing network architectures for stereo matching but hardly achieved both high accuracy and fast inference speed. To ease the workload in network design, neural architecture search (NAS) has been applied with great success to various sparse prediction tasks, such as image classification and object detection. However, existing NAS studies on the dense prediction task, especially stereo matching, still cannot be efficiently and effectively deployed on devices of different computing capabilities. To this end, we propose to train an elastic and accurate network for stereo matching (EASNet) that supports various 3D architectural settings on devices with different computing capabilities. Given the deployment latency constraint on the target device, we can quickly extract a sub-network from the full EASNet without additional training while the accuracy of the sub-network can still be maintained. Extensive experiments show that our EASNet outperforms both state-of-the-art human-designed and NAS-based architectures on Scene Flow and MPI Sintel datasets in terms of model accuracy and inference speed. Particularly, deployed on an inference GPU, EASNet achieves a new SOTA 0.73 EPE on the Scene Flow dataset with 100 ms, which is 4.5$\times$ faster than LEAStereo with a better quality model.