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

=== Assistant: Ran it in my environment g++ -O3 -marchnative -ffast-math, float, measuring: ===
Ran it in my environment (g++ -O3 -march=native -ffast-math, float), measuring:
* runtime: ns/call (median over 9 runs, 50k points)
* accuracy: absolute distance error vs a robust double-precision reference (max / avg / p99), plus maxEqErr = |(x/a)^2+(y/b)^2-1| for the returned point.

===== Moderate (150, 100) =====

| method             | ns/call | max dist err | avg dist err | p99 dist err | maxEqErr  |
| ------------------ | ------- | ------------ | ------------ | ------------ | --------- |
| yours (hybrid 2+1) | 56.295  | 1.080e-02    | 1.209e-05    | 3.053e-05    | 4.520e-07 |
| yours − 3rd step   | 38.502  | 1.879e-01    | 2.480e-04    | 1.169e-03    | 4.737e-07 |
| faded2 (opt)       | 38.899  | 1.879e-01    | 2.480e-04    | 1.169e-03    | 4.737e-07 |
| faded3 (opt)       | 68.299  | 8.955e-03    | 1.152e-05    | 3.146e-05    | 4.419e-07 |
| claude2 (rot)      | 36.109  | 5.183e+01    | 9.692e-01    | 3.275e+01    | 4.495e-07 |
| claude3 (rot)      | 49.230  | 4.905e+01    | 9.457e-01    | 3.277e+01    | 4.370e-07 |

===== High ecc (200, 50) =====

| method             | ns/call | max dist err | avg dist err | p99 dist err | maxEqErr  |
| ------------------ | ------- | ------------ | ------------ | ------------ | --------- |
| yours (hybrid 2+1) | 59.147  | 5.636e-02    | 2.207e-05    | 1.830e-04    | 4.311e-07 |
| yours − 3rd step   | 38.197  | 5.382e-01    | 9.483e-03    | 2.230e-01    | 4.467e-07 |
| faded2 (opt)       | 39.917  | 5.382e-01    | 9.483e-03    | 2.230e-01    | 4.467e-07 |
| faded3 (opt)       | 74.481  | 3.022e-02    | 8.965e-05    | 2.498e-03    | 4.407e-07 |
| claude2 (rot)      | 33.235  | 1.357e+02    | 7.134e+00    | 1.224e+02    | 5.302e-07 |
| claude3 (rot)      | 49.514  | 1.375e+02    | 6.699e+00    | 1.232e+02    | 4.801e-07 |

===== Extreme (100, 10) =====

| method             | ns/call | max dist err | avg dist err | p99 dist err | maxEqErr  |
| ------------------ | ------- | ------------ | ------------ | ------------ | --------- |
| yours (hybrid 2+1) | 58.911  | 1.057e-01    | 2.983e-05    | 3.744e-05    | 4.497e-07 |
| yours − 3rd step   | 38.725  | 5.581e-01    | 1.797e-03    | 4.816e-02    | 4.638e-07 |
| faded2 (opt)       | 40.019  | 5.581e-01    | 1.797e-03    | 4.816e-02    | 4.638e-07 |
| faded3 (opt)       | 71.495  | 1.985e-02    | 3.397e-05    | 7.202e-04    | 4.628e-07 |
| claude2 (rot)      | 33.035  | 8.500e+01    | 5.297e+00    | 7.538e+01    | 5.413e-07 |
| claude3 (rot)      | 49.750  | 8.260e+01    | 4.858e+00    | 7.578e+01    | 4.879e-07 |

===== Circle (100, 100) =====

| method             | ns/call | max dist err | avg dist err | p99 dist err | maxEqErr  |
| ------------------ | ------- | ------------ | ------------ | ------------ | --------- |
| yours (hybrid 2+1) | 55.238  | 3.412e-05    | 5.876e-06    | 1.904e-05    | 4.422e-07 |
| yours − 3rd step   | 39.317  | 3.146e-05    | 3.811e-06    | 1.570e-05    | 3.235e-07 |
| faded2 (opt)       | 39.397  | 3.146e-05    | 3.811e-06    | 1.570e-05    | 3.235e-07 |
| faded3 (opt)       | 70.926  | 3.133e-05    | 4.978e-06    | 1.678e-05    | 3.460e-07 |
| claude2 (rot)      | 34.716  | 3.412e-05    | 5.096e-06    | 1.705e-05    | 4.538e-07 |
| claude3 (rot)      | 49.998  | 3.412e-05    | 5.313e-06    | 1.706e-05    | 4.521e-07 |

Notes you’ll notice from the table:
* <code>yours − 3rd step</code> and <code>faded2 (opt)</code> are the same algorithmic idea (2 iterations of the optimized curvature update), hence the near-identical stats.
* <code>claude2/claude3</code> match the ellipse equation (low maxEqErr) but are not the closest point for non-circles (huge distance error vs robust reference).

If you want to rerun this on your target machine:
Download the benchmark I used<ref>{{cite web|title=Download the benchmark I used|url=https://chatgpt.com/share/Download%20the%20benchmark%20I%20used|publisher=Download the benchmark I used|access-date=2026-01-04}}</ref>