Packing is an essential loop optimization for handcrafting a high-performance General Matrix Multiplication (GEMM). Packing copies a non-contiguous block of data to a contiguous block to reduce the number of TLB entries required to access it, avoiding expensive TLB misses. When copying data, packing can rearrange elements of the block to decrease the stride between consecutive accesses, improving spatial locality. Until now the use of packing has been limited to handcrafted GEMM implementations and to auto-tuning techniques. Existing loop optimizers, such as Polly and Pluto, either only apply packing to GEMM computations (Polly), or not at all (Pluto). This work proposes GPAT, a generalized packing analysis and code transformation that applies packing, when beneficial, to a generic input loop nest. GPAT is implemented in the Affine dialect of MLIR and evaluated on Polybench/C. GPAT applies packing to benchmarks beyond GEMM and obtains significant speedup compared to current loop optimizers that do not apply packing.