Mixed-precision algorithms exploit reduced-precision hardware, however computing with the same matrix at distinct precisions typically requires duplicate storage, repeated conversions, or operation-emulation. We introduce TOMOS, a composite-precision matrix format that provides implicit 32-bit and 16-bit access from a single representation. Each FP32 entry is split into a 16-bit primary word for low-precision computation and a complementary word that enables exact FP32 reconstruction. The encoding scheme improves numerical quality of the low-precision view through subnormal representations, and allows exact recovery of high-precision inputs. TOMOS uses FP32-sized storage, with data movement scaling with compute precision. Optimized GEMM and GEMV kernels achieve throughput comparable to vendor libraries on NVIDIA A100 GPUs, while a mixed-precision least-squares solver achieves up to 1.6x speedup over FP32 without redundant matrix allocations.

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