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

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The results are very clear now:

With -ffast-math: 4x speedup for curvature method Without -ffast-math: 2.6-3x speedup for curvature method

The circle case is interesting - Newton converges very fast for circles (1-2 iterations), so it's nearly tied there.

Let me also create a Rust version to verify these results with another compiler.

Excellent! Let me also create a Rust version for cross-validation:

Good, I've given the user the files. Let me now provide a summary of the results.
C++ Benchmark Results

The results are decisive. 0xfaded's curvature method is 3-4x faster than Newton's method in compiled C++:
Configuration	Curvature	Newton	Speedup
Moderate eccentricity	43 ns	173 ns	4.0x
High eccentricity	44 ns	192 ns	4.4x
Extreme eccentricity	47 ns	194 ns	4.1x
Near-degenerate	42 ns	193 ns	4.6x
Circle	44 ns	70 ns	1.6x
Why It's Faster

The key is avoiding trig functions in the inner loop:
Operation	CPU Cycles
sqrt	~10-15
sin/cos	~50-100+
atan2	~100-150+

    Curvature: 3 iterations × 3 sqrt = 9 sqrt calls
    Newton: 1 atan2 + ~5 iterations × 2 sin/cos = 11+ trig calls

Even without -ffast-math, the curvature method is still 2.6-3x faster.
The Circle Edge Case

For circles (a=b), Newton converges in just 1-2 iterations because the initial atan2 guess is perfect. That's why the speedup drops to 1.6x for that case.
Accuracy

Both achieve identical machine precision (~4×10⁻¹⁶ deviation from ellipse).

The C++ and Rust source files are included if you want to verify on your own machine.
Benchmark results
Document · MD 
Ellipse benchmark
CPP 
Ellipse benchmark
RS 

how does the speedup compare against the python benchmark
8:04 PM

The user wants me to compare the Python benchmark results against the C++ benchmark results. Let me look back at what we found:

Python Results (scalar):

    Curvature was actually slower than Newton in scalar Python
    Newton was about 1.2x faster than curvature in pure Python scalar code

Python Results (vectorized NumPy):

    Curvature was 2-4x faster than Newton when vectorized

C++ Results:

    Curvature was 3-4x faster than Newton consistently

So the interesting comparison is:

    In scalar Python, Newton wins slightly (Python's math.sin/cos are highly optimized C)
    In vectorized Python (NumPy), Curvature wins by 2-4x
    In C++, Curvature wins by 3-4x

The Python scalar result is misleading because Python's interpreter overhead dominates, and the C math library's trig functions are very optimized. The "real" performance difference shows up in compiled code or vectorized operations.

Great question - the comparison reveals something important about microbenchmarking:
Python vs C++ Speedup Comparison
Scenario	Curvature vs Newton
Python scalar	Newton wins ~1.2x
Python vectorized (NumPy)	Curvature wins 2-4x
C++ (-O3 -ffast-math)	Curvature wins 3-4x
C++ (-O3, strict IEEE)	Curvature wins 2.6-3x
Why Python Scalar Was Misleading

In scalar Python, Newton appeared faster because:

    Python's math.sin/math.cos are C library calls - highly optimized, and the Python→C call overhead is the same whether you call sqrt or sin
    Interpreter overhead dominates - the Python bytecode dispatch cost dwarfs the actual math operations
    Newton does fewer iterations - 4-5 vs fixed 3, so fewer Python loop iterations

The Real Picture

                    Python Scalar    Python NumPy    C++ Compiled
                    -------------    ------------    ------------
Curvature method:      6.4 μs          1.2 ms          43 ns
Newton method:         5.2 μs          3.7 ms         173 ns
                         ↓               ↓               ↓
                    Newton wins      Curv wins       Curv wins
                       1.2x            3.0x            4.0x

The C++ results are the ground truth for algorithm comparison. Python scalar benchmarks are measuring Python overhead, not the algorithms themselves.
Key Takeaway

If you're benchmarking algorithms in Python, either:

    Use NumPy/vectorized operations (which bypass the interpreter)
    Or use compiled code (C/C++/Rust)

Scalar Python microbenchmarks can give you the opposite conclusion from reality, as we saw here.

Take the c++ benchmarks we have, and compare against these . Models: C, G, O: 

C: 
Great question! Here are several optimizations for game use:

#include <cmath>

struct Ellipse {
    float cx, cy;
    float a, b;
    // Precomputed values (set once when orbit is defined)
    float a2, b2;        // a², b²
    float ab_diff;       // b² - a²
    float inv_a, inv_b;  // 1/a, 1/b
    
    void precompute() {
        a2 = a * a;
        b2 = b * b;
        ab_diff = b2 - a2;
        inv_a = 1.0f / a;
        inv_b = 1.0f / b;
    }
};

// Optimized version: ~3-4x faster
float distanceToEllipseFast(const Ellipse& e, float px, float py) {
    // Work in first quadrant
    float qx = std::abs(px - e.cx);
    float qy = std::abs(py - e.cy);
    
    // Better initial guess using normalized coordinates
    float nx = qx * e.inv_a;
    float ny = qy * e.inv_b;
    float inv_len = 1.0f / std::sqrt(nx '' nx + ny '' ny + 1e-10f);
    float cosT = nx * inv_len;
    float sinT = ny * inv_len;
    
    // Fixed 4 iterations, no branches, no trig after init
    // Use sin²+cos²=1 to update without calling sin/cos
    for (int i = 0; i < 4; ++i) {
        float ex = e.a * cosT;
        float ey = e.b * sinT;
        
        float f1 = e.ab_diff '' sinT '' cosT + qx '' sinT '' e.a - qy '' cosT '' e.b;
        float f2 = e.ab_diff '' (cosT '' cosT - sinT '' sinT) + qx '' cosT '' e.a + qy '' sinT * e.b;
        
        float dt = f1 / f2;
        
        // Update sin/cos using small angle approximation + renormalization
        // cos(t-dt) ≈ cos(t) + dt''sin(t), sin(t-dt) ≈ sin(t) - dt''cos(t)
        float newCos = cosT + dt * sinT;
        float newSin = sinT - dt * cosT;
        
        // Renormalize to stay on unit circle
        inv_len = 1.0f / std::sqrt(newCos '' newCos + newSin '' newSin);
        cosT = newCos * inv_len;
        sinT = newSin * inv_len;
    }
    
    float dx = e.a * cosT - qx;
    float dy = e.b * sinT - qy;
    return std::sqrt(dx '' dx + dy '' dy);
}

// Even faster: if you just need "inside/outside orbit" checks
float signedDistanceToEllipseCheap(const Ellipse& e, float px, float py) {
    float dx = (px - e.cx) * e.inv_a;
    float dy = (py - e.cy) * e.inv_b;
    float d = std::sqrt(dx '' dx + dy '' dy);
    // Approximate distance (exact on circle, approximate on ellipse)
    return (d - 1.0f) * std::min(e.a, e.b);
}

Key optimizations:
# No trig in the loop — update sin/cos via rotation + renormalization
# Fixed iteration count — no branches, predictable for CPU pipeline
# Precomputed constants — avoid repeated division
# float instead of double — 2x faster, plenty precise for games
# Cheap fallback — if you just need approximate distance or inside/outside tests
If you need batch processing (many points):

#include <immintrin.h>

// Process 8 points at once with AVX
void distanceToEllipseSIMD(const Ellipse& e, 
                            const float'' px, const float'' py,
                            float* out, int count) {
    __m256 va = _mm256_set1_ps(e.a);
    __m256 vb = _mm256_set1_ps(e.b);
    __m256 vcx = _mm256_set1_ps(e.cx);
    __m256 vcy = _mm256_set1_ps(e.cy);
    // ... vectorized version of the above
}

For most games, the 4-iteration float version is accurate to sub-pixel precision and should handle tens of thousands of queries per frame without issue.

G:  paste1

O: paste2
8:08 PM

The user wants me to compare the curvature-based method (0xfaded's) against these game-optimized Newton variants. Let me create a comprehensive benchmark that includes:

    Original curvature method (0xfaded)
    Original Newton method
    Model C's optimized Newton (sin/cos rotation trick)
    Model G's fast Newton (Eberly-style t equation)
    Model O's fast Newton (similar Eberly-style)

The key differences to test:

    Curvature: 3 iterations, no trig in loop, uses sqrt
    Newton variants: 2-4 iterations, various optimizations

Let me write a comprehensive C++ benchmark.

Let me create a comprehensive benchmark comparing all these approaches:

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

Output