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

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Methods ranked by typical performance:
# Curvature (0xfaded)   - ~20-25 ns - NO TRIG, 3 sqrt/iter, fixed 3 iters
# Model G (Eberly 2)    - ~25-35 ns - NO TRIG, 2 div/iter, but less accurate  
# Model O (Eberly 2)    - ~30-40 ns - NO TRIG, better init, still 2 iter
# Model O (Eberly 3)    - ~35-45 ns - NO TRIG, 3 iter for better accuracy
# Model C (rotation)    - ~35-50 ns - NO TRIG in loop, 4 iters, sqrt/iter
# Newton Standard       - ~80-120ns - TRIG every iteration

Key insight: 
* Curvature method uses 3 sqrt per iteration = 9 sqrt total
* Eberly methods use 2-3 div per iteration = 4-9 div total
* Both avoid trig, but curvature has simpler iteration structure

For GAMES: Model G/O with 2 iterations is often "good enough" and fastest.
For ACCURACY: Curvature 3-iter or Model O 3-iter are best choices.

Whoa! The Model G and Model O methods have terrible accuracy (errors in the millions!). That means there's a bug in those implementations or the algorithms don't work correctly for all cases. Let me look at the accuracy numbers:

    Curvature: 4.77e-07 (good - float precision)
    Newton Standard: 1.19e-07 (good)
    Model G (2): 1.31e+06 (TERRIBLE - completely wrong!)
    Model O (2): 6.65e+10 (TERRIBLE - completely wrong!)
    Model O (3): 2.96e+10 (TERRIBLE - completely wrong!)

The Eberly t-equation methods are completely broken! The Model C (rotation trick) doesn't have accuracy listed because it needs special handling. Let me fix this benchmark to properly compare only working methods.

Looking at the Eberly method more carefully, the issue is that t=0 initialization doesn't work for all points. The method needs more careful handling. Let me fix the implementations.

Actually, looking at the original code from Models G and O, there are issues:

    Model G starts t=0 which doesn't converge for many cases
    Model O has issues with inside/outside handling

Let me create a proper version of the Eberly method that actually works, and then do a fair comparison.

Whoa! The Eberly-style methods (Model G/O) have catastrophic accuracy failures - errors in the millions! Those implementations are broken. Let me fix them and re-run:

bash
cd /home/claude && g++ -O3 -march=native -ffast-math -std=c++17 ellipse_benchmark_fixed.cpp -o bench_fixed -lm && ./bench_fixed

Output