Progressive Real-Time Computation

We present a progressive computation method for square roots that rapidly decomposes a number into integer and rational components, delivering guaranteed five-digit precision after the decimal point without correction tables. The first stage yields a coarse estimate with under one percent error in just 3–17 clock cycles, enabling the system to act before computation completes. Full evaluation fits within 7 cycles when leveraging hardware acceleration and a compact reciprocal table. When summing multiple roots, layers are processed in parallel: the first-stage result becomes available within a few cycles rather than after all computations finish. The method provides deterministic latency with jitter below 5%, critical for real-time systems, and relies exclusively on integer arithmetic—eliminating dependencies on floating-point units or memory cache behavior.