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

=== Assistant: Yeah — I double-checked it, and Claude’s “faster” result is an artifact of the reference being wrong, not an actually-b… ===
Yeah — I double-checked it, and Claude’s “faster” result is an artifact of the reference being wrong, not an actually-better closest-point solver.

===== I took the same set of methods and timing harness style, but replaced the “reference” with a robust closest-point reference (Eberly’s monotonic t-equation with bracketing), which has a unique root → unique closest point for any ellipse/point. =====

Claude’s file uses as “reference” a Newton solve on the angle equation starting from atan2(a|y|, b|x|) — that can converge to the wrong stationary point (still on the ellipse, but not the closest), especially for interior points and high eccentricity.

===== With the robust reference, claude_rotation is fast but wildly incorrect for real ellipses (except the circle case): =====

Example (this environment, -O3 -march=native -ffast-math, N=20k):

Moderate (150,100):
* claude_rot(2) ≈ 43.9 ns/call, but p99 distance error ~31 and max error ~43 (not closest point).
* faded_opt(2) ≈ 50.6 ns/call, p99 ~1.2e-3, max ~0.20 (close).
* faded_opt(3) ≈ 84.1 ns/call, p99 ~3e-5, max ~1.3e-2 (very close).

High ecc (200,50):
* claude_rot(2) ≈ 39.1 ns/call, but p99 ~120 (!!)
* faded_opt(3) ≈ 91.0 ns/call, p99 ~2.4e-3

Circle (100,100):
* claude_rot(2) is fine (projection is trivial-ish), errors ~1e-5.

===== Because his “reference” is essentially the same Newton basin as claude_rotation, so they often converge to the same wrong stationary point and therefore agree—making the reported “distance error vs reference” look small even when both are wrong. =====

===== - Yes, Claude found something faster in ns/call. =====
* No, it’s not a correct closest-point solver for ellipses (it’s returning the wrong stationary point a lot).
* If you want “fast and correct”, stick with faded_opt(3) (or a carefully validated hybrid), and validate against a monotonic/bracketed reference like Eberly.

If you want, I can paste the exact double-check harness I ran (the one producing those numbers) so you can compile it on your target machine.