Error Analysis for Fused Floating-point Square-root and Division based on Goldschmidt Algorithm

This paper presents an error analysis method for floating-point square-root and division based on Goldschmidt algorithm. The proposed analysis is sufficiently concise to guide circuit design and leads to intermediate results for state-of-the-art word length reduction. This error analysis method is applicable to both division and square-root and can help designers achieve a high degree of resource reuse for these two operations. To validate our method, we have designed a fused square-root and division module for IEEE 754 single precision and compare the theoretical error bound with the simulated error bound under 100 trillion random tests. The module not only excels among existing floating-point units, but also achieves a high degree of resource reuse for division and square-root.