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

=== User: no I'm not looking for  30ns or less, this would run on a different environment so you'd need to run the whole benchmar… ===
no I'm not looking for  30ns or less, this would run on a different environment so you'd need to run the whole benchmark and make yourself a more optimized more efficient solution. that last run was on claude's container. 

// ellipse_benchmark_v2.cpp
// Updated with 0xfaded's suggestions:
// 1. Combine r/q into single sqrt
// 2. Remove the q < 1e-10 guard
// 3. Add extreme eccentricity cases (a=1, b=100)
// 4. Try std::hypot
//
// g++ -O3 -march=native -ffast-math -std=c++17 ellipse_benchmark_v2.cpp -o bench_v2 -lm

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

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

// ============================================================================
// ORIGINAL: 0xfaded's curvature method (as benchmarked before)
// ============================================================================
inline Point curvature_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 < 3; 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)};
}

// ============================================================================
// OPTIMIZED: Combined r/q into single sqrt, removed guard
// ============================================================================
inline Point curvature_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 < 3; 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;
        
        // Combined: r/q in one sqrt
        float rq = std::sqrt((rx''rx + ry''ry) / (qx''qx + qy''qy));
        
        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)};
}

// ============================================================================
// HYPOT version: using std::hypot
// ============================================================================
inline Point curvature_hypot(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 < 3; 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::hypot(rx, ry);
        float q = std::hypot(qx, qy);
        
        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::hypot(tx, ty);
        tx /= t; ty /= t;
    }
    return {std::copysign(a''tx, px), std::copysign(b''ty, py)};
}

// ============================================================================
// Claude's rotation trick (for comparison)
// ============================================================================
inline Point newton_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 < 4; 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)};
}

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

template<typename F>
double bench(F fn, float a, float b, const std::vector<Point>& pts, int runs = 10) {
    // Warmup
    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 < runs; 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];
}

float accuracy(float a, float b, const std::vector<Point>& pts, Point (*fn)(float,float,float,float)) {
    float maxe = 0;
    for (auto& p : pts) {
        Point r = fn(a, b, p.x, p.y);
        float e = std::fabs((r.x/a)''(r.x/a) + (r.y/b)''(r.y/b) - 1.f);
        if (!std::isnan(e) && !std::isinf(e)) maxe = std::fmax(maxe, e);
    }
    return maxe;
}

int main() {
    printf("=========================================================\n");
    printf("  BENCHMARK v2: With 0xfaded's suggested optimizations\n");
    printf("=========================================================\n\n");
    
    std::mt19937 rng(42);
    std::uniform_real_distribution<float> angle(0, 2*M_PI);
    std::uniform_real_distribution<float> radius(0.5f, 2.5f);
    
    const int N = 50000;
    
    struct Cfg { float a, b; const char* name; } cfgs[] = {
        {150, 100, "Moderate (150, 100)"},
        {200, 50,  "High ecc (200, 50)"},
        {100, 10,  "Very high (100, 10)"},
        {1, 100,   "Extreme (1, 100)"},      // 0xfaded's suggestion
        {100, 1,   "Extreme (100, 1)"},      // 0xfaded's suggestion
        {1, 1000,  "Extreme (1, 1000)"},     // Even more extreme
        {100, 100, "Circle (100, 100)"},
    };
    
    for (auto& cfg : cfgs) {
        std::vector<Point> 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)};
        }
        
        double t_orig = bench(curvature_original, cfg.a, cfg.b, pts);
        double t_opt = bench(curvature_optimized, cfg.a, cfg.b, pts);
        double t_hyp = bench(curvature_hypot, cfg.a, cfg.b, pts);
        double t_rot = bench(newton_rotation, cfg.a, cfg.b, pts);
        
        printf("%-25s\n", cfg.name);
        printf("  original     %6.1f ns   acc: %.1e\n", t_orig, accuracy(cfg.a, cfg.b, pts, curvature_original));
        printf("  optimized    %6.1f ns   acc: %.1e  (%.0f%% of original)\n", 
               t_opt, accuracy(cfg.a, cfg.b, pts, curvature_optimized), 100*t_opt/t_orig);
        printf("  hypot        %6.1f ns   acc: %.1e  (%.0f%% of original)\n", 
               t_hyp, accuracy(cfg.a, cfg.b, pts, curvature_hypot), 100*t_hyp/t_orig);
        printf("  rotation     %6.1f ns   acc: %.1e\n", t_rot, accuracy(cfg.a, cfg.b, pts, newton_rotation));
        printf("\n");
    }
    
    printf("=========================================================\n");
    printf("  SUMMARY\n");
    printf("=========================================================\n");
    printf("  original  = 0xfaded's code as I benchmarked it\n");
    printf("  optimized = combined r/q into single sqrt, no guard\n");
    printf("  hypot     = using std::hypot instead of manual sqrt\n");
    printf("  rotation  = Claude's sin/cos rotation trick\n");
    printf("=========================================================\n");
    
    return 0;
}