Providing Accuracy Guarantees for the Fixed-Point Emulation of Matrix Multiplication

Fixed-point methods such as Ozaki-II emulate double-precision matrix multiplication (GEMM) on fast low-precision hardware. Software that chooses the mantissa bit count automatically is numerically safe when the emulated product keeps the error guarantee of the floating-point GEMM it replaces. We analyze the worst-case and average-case error of fixed-point and floating-point dot products, define exponent span capacities that determine the extra mantissa bits an input requires, and approximate them with a low-precision GEMM. We prove that 53 bits plus such a capacity and a small constant preserve the classical worst-case error bound for a floating-point dot product, and we demonstrate this guarantee in software on several test cases. In a developer build of cuBLAS on an NVIDIA B200, each implemented bit-count rule earns grade~A on Test~2c of ``Grading the BLAS’' at every size and exponent range tested, falling back to double precision at the widest ranges; in exact arithmetic, the bound holds on every adversarial and random input tested.