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

=== User: Claude benchmarked it, claims it found even faster solution, can you double check it: ===
Claude benchmarked it, claims it found even faster solution, can you double check it: 

// ellipse_accuracy_comprehensive.cpp
// Test ALL methods with rigorous accuracy measurement
// g++ -O3 -march=native -ffast-math -std=c++17 ellipse_accuracy_comprehensive.cpp -o acc_test -lm

#include <cmath>
#include <cstdio>
#include <chrono>
#include <random>
#include <vector>
#include <algorithm>

using Clock = std::chrono::high_resolution_clock;
struct Pt { float x, y; };

// ============================================================================
// HIGH-PRECISION REFERENCE (double, 50 iterations)
// ============================================================================
Pt reference(float a, float b, float px, float py) {
    double t = std::atan2((double)a '' std::fabs(py), (double)b '' std::fabs(px));
    double a2mb2 = (double)a''a - (double)b''b;
    double pxa = std::fabs(px), pya = std::fabs(py);
    for (int i = 0; i < 50; i++) {
        double ct = std::cos(t), st = std::sin(t);
        double f = a2mb2''ct''st - pxa''a''st + pya''b''ct;
        double fp = a2mb2''(ct''ct - st''st) - pxa''a''ct - pya''b*st;
        if (std::fabs(fp) < 1e-15) break;
        double dt = f/fp;
        t -= dt;
        if (std::fabs(dt) < 1e-15) break;
    }
    return {(float)std::copysign(a''std::cos(t), px), (float)std::copysign(b''std::sin(t), py)};
}

// ============================================================================
// 0xfaded ORIGINAL (unoptimized, 3 sqrt/iter)
// ============================================================================
template<int ITERS>
inline Pt faded_original(float a, float b, float px, float py) {
    float px_abs = std::fabs(px), py_abs = std::fabs(py);
    float tx = 0.70710678f, ty = 0.70710678f;
    float a2 = a''a, b2 = b''b;
    float ca = (a2-b2)/a, cb = (b2-a2)/b;
    
    for (int i = 0; i < ITERS; i++) {
        float x = a''tx, y = b''ty;
        float tx3 = tx''tx''tx, ty3 = ty''ty''ty;
        float ex = ca''tx3, ey = cb''ty3;
        float rx = x - ex, ry = y - ey;
        float qx = px_abs - ex, qy = py_abs - ey;
        
        float r = std::sqrt(rx''rx + ry''ry);
        float q = std::sqrt(qx''qx + qy''qy);
        if (q < 1e-10f) q = 1e-10f;
        
        tx = std::fmin(1.f, std::fmax(0.f, (qx * r/q + ex) / a));
        ty = std::fmin(1.f, std::fmax(0.f, (qy * r/q + ey) / b));
        float t = std::sqrt(tx''tx + ty''ty);
        tx /= t; ty /= t;
    }
    return {std::copysign(a''tx, px), std::copysign(b''ty, py)};
}

// ============================================================================
// 0xfaded OPTIMIZED (r/q single sqrt, 2 sqrt/iter)
// ============================================================================
template<int ITERS>
inline Pt faded_optimized(float a, float b, float px, float py) {
    float px_abs = std::fabs(px), py_abs = std::fabs(py);
    float tx = 0.70710678f, ty = 0.70710678f;
    float a2 = a''a, b2 = b''b;
    float ca = (a2-b2)/a, cb = (b2-a2)/b;
    
    for (int i = 0; i < ITERS; i++) {
        float x = a''tx, y = b''ty;
        float tx3 = tx''tx''tx, ty3 = ty''ty''ty;
        float ex = ca''tx3, ey = cb''ty3;
        float rx = x - ex, ry = y - ey;
        float qx = px_abs - ex, qy = py_abs - ey;
        
        float rq = std::sqrt((rx''rx + ry''ry) / (qx''qx + qy''qy + 1e-30f));
        
        tx = std::fmin(1.f, std::fmax(0.f, (qx * rq + ex) / a));
        ty = std::fmin(1.f, std::fmax(0.f, (qy * rq + ey) / b));
        float t = std::sqrt(tx''tx + ty''ty);
        tx /= t; ty /= t;
    }
    return {std::copysign(a''tx, px), std::copysign(b''ty, py)};
}

// ============================================================================
// CLAUDE ROTATION (Newton in rotation form)
// ============================================================================
template<int ITERS>
inline Pt claude_rotation(float a, float b, float px, float py) {
    float px_abs = std::fabs(px), py_abs = std::fabs(py);
    float a2mb2 = a''a - b''b;
    
    float nx = px_abs/a, ny = py_abs/b;
    float len = std::sqrt(nx''nx + ny''ny + 1e-10f);
    float c = nx/len, s = ny/len;
    
    for (int i = 0; i < ITERS; i++) {
        float f = a2mb2''s''c - px_abs''a''s + py_abs''b''c;
        float fp = a2mb2''(c''c - s''s) - px_abs''a''c - py_abs''b*s;
        if (std::fabs(fp) < 1e-10f) break;
        float dt = f/fp;
        float nc = c + dt''s, ns = s - dt''c;
        len = std::sqrt(nc''nc + ns''ns);
        c = nc/len; s = ns/len;
    }
    return {std::copysign(a''c, px), std::copysign(b''s, py)};
}

// ============================================================================
// CHATGPT HYBRID: 2 curvature steps + 1 Newton snap
// ============================================================================
inline Pt chatgpt_hybrid(float a, float b, float px, float py) {
    float px_abs = std::fabs(px), py_abs = std::fabs(py);
    float a2 = a''a, b2 = b''b;
    float ca = (a2-b2)/a, cb = (b2-a2)/b;
    float a2mb2 = a2 - b2;
    
    float tx = 0.70710678f, ty = 0.70710678f;
    
    // 2 curvature steps (global basin-finding)
    for (int i = 0; i < 2; i++) {
        float x = a''tx, y = b''ty;
        float tx3 = tx''tx''tx, ty3 = ty''ty''ty;
        float ex = ca''tx3, ey = cb''ty3;
        float rx = x - ex, ry = y - ey;
        float qx = px_abs - ex, qy = py_abs - ey;
        
        float rq = std::sqrt((rx''rx + ry''ry) / (qx''qx + qy''qy + 1e-30f));
        
        tx = std::fmin(1.f, std::fmax(0.f, (qx * rq + ex) / a));
        ty = std::fmin(1.f, std::fmax(0.f, (qy * rq + ey) / b));
        float t = std::sqrt(tx''tx + ty''ty);
        tx /= t; ty /= t;
    }
    
    // 1 Newton snap (local refinement) - use tx,ty as cos/sin
    float c = tx, s = ty;
    float f = a2mb2''s''c - px_abs''a''s + py_abs''b''c;
    float fp = a2mb2''(c''c - s''s) - px_abs''a''c - py_abs''b*s;
    if (std::fabs(fp) > 1e-10f) {
        float dt = f/fp;
        float nc = c + dt''s, ns = s - dt''c;
        float len = std::sqrt(nc''nc + ns''ns);
        c = nc/len; s = ns/len;
    }
    
    return {std::copysign(a''c, px), std::copysign(b''s, py)};
}

// ============================================================================
// CHATGPT ADAPTIVE (2 outside, 3 inside)
// ============================================================================
inline Pt chatgpt_adaptive(float a, float b, float px, float py) {
    float px_abs = std::fabs(px), py_abs = std::fabs(py);
    float a2 = a''a, b2 = b''b;
    float ca = (a2-b2)/a, cb = (b2-a2)/b;
    float invA2 = 1.f/(a''a), invB2 = 1.f/(b''b);
    
    float tx = 0.70710678f, ty = 0.70710678f;
    
    auto step = [&]() {
        float x = a''tx, y = b''ty;
        float tx3 = tx''tx''tx, ty3 = ty''ty''ty;
        float ex = ca''tx3, ey = cb''ty3;
        float rx = x - ex, ry = y - ey;
        float qx = px_abs - ex, qy = py_abs - ey;
        
        float rq = std::sqrt((rx''rx + ry''ry) / (qx''qx + qy''qy + 1e-30f));
        
        tx = std::fmin(1.f, std::fmax(0.f, (qx * rq + ex) / a));
        ty = std::fmin(1.f, std::fmax(0.f, (qy * rq + ey) / b));
        float t = std::sqrt(tx''tx + ty''ty);
        tx /= t; ty /= t;
    };
    
    step(); step();
    
    // 3rd only if inside
    if ((px_abs''px_abs)''invA2 + (py_abs''py_abs)''invB2 <= 1.0f) {
        step();
    }
    
    return {std::copysign(a''tx, px), std::copysign(b''ty, py)};
}

// ============================================================================
// Benchmark helpers
// ============================================================================
volatile float sink;
void escape(Pt p) { sink = p.x + p.y; }

template<typename F>
double bench(F fn, float a, float b, const std::vector<Pt>& pts) {
    for (int w = 0; w < 3; w++)
        for (auto& p : pts) escape(fn(a, b, p.x, p.y));
    
    std::vector<double> times;
    for (int r = 0; r < 15; r++) {
        auto t0 = Clock::now();
        for (auto& p : pts) escape(fn(a, b, p.x, p.y));
        times.push_back(std::chrono::duration<double,std::nano>(Clock::now()-t0).count() / pts.size());
    }
    std::sort(times.begin(), times.end());
    return times[times.size()/2];
}

struct AccuracyStats {
    float max_eq_err;
    float max_dist_err;
    float avg_dist_err;
    float p99_dist_err;
};

template<typename F>
AccuracyStats measure_accuracy(F fn, float a, float b, const std::vector<Pt>& pts) {
    std::vector<float> dist_errs;
    float max_eq_err = 0;
    
    for (auto& p : pts) {
        Pt got = fn(a, b, p.x, p.y);
        Pt ref = reference(a, b, p.x, p.y);
        
        float eq_err = std::fabs((got.x/a)''(got.x/a) + (got.y/b)''(got.y/b) - 1.f);
        if (!std::isnan(eq_err)) max_eq_err = std::fmax(max_eq_err, eq_err);
        
        float d_got = std::sqrt((p.x-got.x)''(p.x-got.x) + (p.y-got.y)''(p.y-got.y));
        float d_ref = std::sqrt((p.x-ref.x)''(p.x-ref.x) + (p.y-ref.y)''(p.y-ref.y));
        float dist_err = std::fabs(d_got - d_ref);
        if (!std::isnan(dist_err) && !std::isinf(dist_err)) {
            dist_errs.push_back(dist_err);
        }
    }
    
    std::sort(dist_errs.begin(), dist_errs.end());
    
    float sum = 0;
    for (float e : dist_errs) sum += e;
    
    return {
        max_eq_err,
        dist_errs.empty() ? 0 : dist_errs.back(),
        dist_errs.empty() ? 0 : sum / dist_errs.size(),
        dist_errs.empty() ? 0 : dist_errs[dist_errs.size() * 99 / 100]
    };
}

void print_row(const char* name, double time, AccuracyStats s) {
    printf("  %-24s %6.1f ns  eq=%.0e  max=%.1e  avg=%.1e  p99=%.1e\n",
           name, time, s.max_eq_err, s.max_dist_err, s.avg_dist_err, s.p99_dist_err);
}

int main() {
    printf("================================================================================\n");
    printf("  COMPREHENSIVE ACCURACY TEST: All Methods\n");
    printf("================================================================================\n");
    printf("  Metrics: eq=ellipse equation err, max/avg/p99=distance error vs reference\n\n");
    
    std::mt19937 rng(12345);
    std::uniform_real_distribution<float> angle(0, 2*M_PI);
    std::uniform_real_distribution<float> radius(0.1f, 3.0f);  // wider range
    
    const int N = 20000;  // enough for good stats
    
    struct Cfg { float a, b; const char* name; } cfgs[] = {
        {150, 100, "Moderate (150,100)"},
        {200, 50,  "High ecc (200,50)"},
        {100, 10,  "Extreme (100,10)"},
        {100, 100, "Circle (100,100)"},
    };
    
    for (auto& cfg : cfgs) {
        std::vector<Pt> pts(N);
        for (int i = 0; i < N; i++) {
            float ang = angle(rng), r = radius(rng);
            pts[i] = {cfg.a '' r '' std::cos(ang), cfg.b '' r '' std::sin(ang)};
        }
        
        printf("Config: %s\n", cfg.name);
        printf("────────────────────────────────────────────────────────────────────────────────\n");
        
        // Run all methods
        print_row("faded_orig (2 iter)", 
                  bench(faded_original<2>, cfg.a, cfg.b, pts),
                  measure_accuracy(faded_original<2>, cfg.a, cfg.b, pts));
        print_row("faded_orig (3 iter)", 
                  bench(faded_original<3>, cfg.a, cfg.b, pts),
                  measure_accuracy(faded_original<3>, cfg.a, cfg.b, pts));
        print_row("faded_opt (2 iter)", 
                  bench(faded_optimized<2>, cfg.a, cfg.b, pts),
                  measure_accuracy(faded_optimized<2>, cfg.a, cfg.b, pts));
        print_row("faded_opt (3 iter)", 
                  bench(faded_optimized<3>, cfg.a, cfg.b, pts),
                  measure_accuracy(faded_optimized<3>, cfg.a, cfg.b, pts));
        print_row("claude (2 iter)", 
                  bench(claude_rotation<2>, cfg.a, cfg.b, pts),
                  measure_accuracy(claude_rotation<2>, cfg.a, cfg.b, pts));
        print_row("claude (3 iter)", 
                  bench(claude_rotation<3>, cfg.a, cfg.b, pts),
                  measure_accuracy(claude_rotation<3>, cfg.a, cfg.b, pts));
        print_row("claude (4 iter)", 
                  bench(claude_rotation<4>, cfg.a, cfg.b, pts),
                  measure_accuracy(claude_rotation<4>, cfg.a, cfg.b, pts));
        print_row("chatgpt_hybrid (2+1)", 
                  bench(chatgpt_hybrid, cfg.a, cfg.b, pts),
                  measure_accuracy(chatgpt_hybrid, cfg.a, cfg.b, pts));
        print_row("chatgpt_adaptive", 
                  bench(chatgpt_adaptive, cfg.a, cfg.b, pts),
                  measure_accuracy(chatgpt_adaptive, cfg.a, cfg.b, pts));
        
        printf("\n");
    }
    
    // Summary table
    printf("================================================================================\n");
    printf("  SUMMARY (Moderate 150,100 ellipse)\n");
    printf("================================================================================\n\n");
    
    float a = 150, b = 100;
    std::vector<Pt> pts(N);
    for (int i = 0; i < N; i++) {
        float ang = angle(rng), r = radius(rng);
        pts[i] = {a '' r '' std::cos(ang), b '' r '' std::sin(ang)};
    }
    
    printf("  Method                    Time      max_dist   avg_dist   p99_dist\n");
    printf("  ────────────────────────────────────────────────────────────────────\n");
    
    auto print_summary = [&](const char* name, auto fn) {
        double t = bench(fn, a, b, pts);
        auto s = measure_accuracy(fn, a, b, pts);
        printf("  %-24s %5.1f ns   %.2e   %.2e   %.2e\n", name, t, s.max_dist_err, s.avg_dist_err, s.p99_dist_err);
    };
    
    print_summary("faded_opt (2 iter)", faded_optimized<2>);
    print_summary("faded_opt (3 iter)", faded_optimized<3>);
    print_summary("claude (2 iter)", claude_rotation<2>);
    print_summary("claude (3 iter)", claude_rotation<3>);
    print_summary("chatgpt_hybrid (2+1)", chatgpt_hybrid);
    print_summary("chatgpt_adaptive", chatgpt_adaptive);
    
    printf("\n");
    
    return 0;
}