Average Error: 32.2 → 0.7
Time: 20.6s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\left(\left(\left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \frac{x}{y}\right) \cdot \sqrt[3]{\frac{x}{y}} + \frac{z}{t} \cdot \frac{z}{t}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\left(\left(\left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \frac{x}{y}\right) \cdot \sqrt[3]{\frac{x}{y}} + \frac{z}{t} \cdot \frac{z}{t}
double f(double x, double y, double z, double t) {
        double r29135665 = x;
        double r29135666 = r29135665 * r29135665;
        double r29135667 = y;
        double r29135668 = r29135667 * r29135667;
        double r29135669 = r29135666 / r29135668;
        double r29135670 = z;
        double r29135671 = r29135670 * r29135670;
        double r29135672 = t;
        double r29135673 = r29135672 * r29135672;
        double r29135674 = r29135671 / r29135673;
        double r29135675 = r29135669 + r29135674;
        return r29135675;
}
↓
double f(double x, double y, double z, double t) {
        double r29135676 = 1.0;
        double r29135677 = y;
        double r29135678 = r29135676 / r29135677;
        double r29135679 = cbrt(r29135678);
        double r29135680 = x;
        double r29135681 = cbrt(r29135680);
        double r29135682 = r29135679 * r29135681;
        double r29135683 = r29135680 / r29135677;
        double r29135684 = cbrt(r29135683);
        double r29135685 = r29135682 * r29135684;
        double r29135686 = r29135685 * r29135683;
        double r29135687 = r29135686 * r29135684;
        double r29135688 = z;
        double r29135689 = t;
        double r29135690 = r29135688 / r29135689;
        double r29135691 = r29135690 * r29135690;
        double r29135692 = r29135687 + r29135691;
        return r29135692;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.2
Target0.4
Herbie0.7
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 32.2

    \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right)}\]
  3. Using strategy rm

  4. Applied fma-udef0.4

    \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t} + \frac{x}{y} \cdot \frac{x}{y}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.8

    \[\leadsto \frac{z}{t} \cdot \frac{z}{t} + \frac{x}{y} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \sqrt[3]{\frac{x}{y}}\right)}\]

  7. Applied associate-*r*0.8

    \[\leadsto \frac{z}{t} \cdot \frac{z}{t} + \color{blue}{\left(\frac{x}{y} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)\right) \cdot \sqrt[3]{\frac{x}{y}}}\]
  8. Using strategy rm

  9. Applied div-inv0.8

    \[\leadsto \frac{z}{t} \cdot \frac{z}{t} + \left(\frac{x}{y} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\color{blue}{x \cdot \frac{1}{y}}}\right)\right) \cdot \sqrt[3]{\frac{x}{y}}\]

  10. Applied cbrt-prod0.7

    \[\leadsto \frac{z}{t} \cdot \frac{z}{t} + \left(\frac{x}{y} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{\frac{1}{y}}\right)}\right)\right) \cdot \sqrt[3]{\frac{x}{y}}\]

  11. Final simplification0.7

    \[\leadsto \left(\left(\left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \frac{x}{y}\right) \cdot \sqrt[3]{\frac{x}{y}} + \frac{z}{t} \cdot \frac{z}{t}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"

  :herbie-target
  (+ (pow (/ x y) 2) (pow (/ z t) 2))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))