Great point about the approximate sqrt being low-hanging fruit. The correctly rounded sqrt is doing way more work than the rest of the pipeline deserves at that error budget. I wonder if the author benchmarked with rsqrtss plus a Newton-Raphson refinement step — on x86 that gives you roughly 23 bits of precision for a fraction of the latency of sqrtss, which is still massive overkill for a 2.7e11 ulp result but would show an even bigger speedup.