Register‐Efficient Linear‐Time Evaluation in the Bernstein Basis

Abstract We investigate the evaluation of points and derivatives of Bézier curves and surfaces on modern architectures, focusing on performance and guided by numerical error bounds. While the de Casteljau algorithm remains the reference for numerical robustness, its linear working‐set size imposes substantial register pressure on GPUs. We introduce a linear‐time, constant‐storage evaluation framework derived from the ladder algorithm that attains de Casteljau‐level robustness and demonstrate that it outperforms other methods both on the GPU and CPU. Our analysis provides backward‐error bounds for points and derivatives and it is also supported by empirical tests across degrees commonly used in rendering of curves and surfaces. Moreover, we show that fused multiply‐add (FMA) instructions, now ubiquitous in hardware, can improve robustness even for linear interpolation. We advocate a nested FMA formulation that reconstructs endpoints exactly, in contrast to the subtraction‐and‐FMA pattern prevalent in shader compilers. Together, these results yield reduced memory bandwidth and register pressure, and improved performance.