The diffusion inversion problem seeks to recover the latent generative trajectory of a diffusion model given a real image. Faithful inversion is critical for ensuring consistency in diffusion-based image editing. Prior works formulate this task as a fixed-point problem and solve it using numerical methods. However, achieving both accuracy and efficiency remains challenging, especially for few-step models and novel samples. In this paper, we propose PreciseInv, a general-purpose test-time optimization framework that enables fast and faithful inversion in as few as two inference steps. Unlike root-finding methods, we reformulate inversion as a learning problem and introduce a dynamic programming-inspired strategy to recursively estimate a parameterized sequence of noise embeddings. This design leverages the smoothness of the diffusion latent space for accurate gradient-based optimization and ensures memory efficiency via recursive subproblem construction. We further provide a theoretical analysis of PreciseInv's convergence and derive a provable upper bound on its reconstruction error. Extensive experiments on COCO 2017, DarkFace, and a stylized cartoon dataset show that PreciseInv achieves state-of-the-art performance in both reconstruction quality and inference speed. Improvements are especially notable for few-step models and under distribution shifts. Moreover, precise inversion yields substantial gains in editing consistency for text-driven image manipulation tasks. Code is available at: https://github.com/panda7777777/PreciseInv