Accuracy targets in mixed-precision arithmetic may be unreachable regardless of limb count

Mixed-precision kernels are typically tuned by choosing a format, limb count, and accumulator width, then checking the error after the fact, which leaves unclear which accuracy targets are achievable at all. A paper titled How Many Bits Does a Trusted Result Need? addresses this by measuring the accuracy budget against exact integer and rational arithmetic over a hash-pinned matrix corpus. Its results show that once the accumulator's rounding dominates, extra limbs stop improving accuracy, and the error floor depends on register width rather than the format's mantissa. The reachable error floor scales with the condition number, but the constant of proportionality varies by roughly a factor of three to six across matrices with equal condition numbers. The authors also find that changing only the summation order can shift the error by up to two orders of magnitude, a factor that standard precision budgets do not account for.