Editing Openai/6959ed27-accc-800e-8e87-21aa81e93c07 (section)

==== You already used the “save a sqrt by folding r/q” trick (from 0xfaded’s own HN reply) Hacker News<ref>{{cite web|title=Hacker News|url=https://news.ycombinator.com/item?id=46482345|publisher=Hacker News|access-date=2026-01-04}}</ref>, which was the best micro optimization. ====

But the biggest macro optimization is:
don’t pay for the 3rd iteration unless you need it.

That’s how you get a meaningfully faster implementation without drifting into the “2 iterations but wrong root / wrong stationary point” failure mode that kills Newton-style approaches inside the ellipse.

If you want, I can also give you:
* an AVX2 “8 points at a time” version (throughput monster),
* a rotated-ellipse wrapper (precompute sin/cos once per ellipse),
* a tiny benchmark harness that prints ns/call + max distance error (so you can paste it into HN like you did).