Average Error: 32.2 → 0.9
Time: 24.1s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \left(\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y}}}{\frac{y}{x}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \left(\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y}}}{\frac{y}{x}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)
double f(double x, double y, double z, double t) {
        double r25005680 = x;
        double r25005681 = r25005680 * r25005680;
        double r25005682 = y;
        double r25005683 = r25005682 * r25005682;
        double r25005684 = r25005681 / r25005683;
        double r25005685 = z;
        double r25005686 = r25005685 * r25005685;
        double r25005687 = t;
        double r25005688 = r25005687 * r25005687;
        double r25005689 = r25005686 / r25005688;
        double r25005690 = r25005684 + r25005689;
        return r25005690;
}
↓
double f(double x, double y, double z, double t) {
        double r25005691 = z;
        double r25005692 = t;
        double r25005693 = r25005691 / r25005692;
        double r25005694 = x;
        double r25005695 = cbrt(r25005694);
        double r25005696 = y;
        double r25005697 = cbrt(r25005696);
        double r25005698 = r25005695 / r25005697;
        double r25005699 = r25005696 / r25005694;
        double r25005700 = r25005698 / r25005699;
        double r25005701 = r25005700 * r25005698;
        double r25005702 = r25005698 * r25005701;
        double r25005703 = fma(r25005693, r25005693, r25005702);
        return r25005703;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original32.2
Target0.4
Herbie0.9
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

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 add-cube-cbrt0.8

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

  5. Applied add-cube-cbrt0.9

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

  6. Applied times-frac0.9

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

  7. Applied associate-*r*0.9

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

  8. Simplified0.9

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

  9. Final simplification0.9

    \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \left(\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{y}}}{\frac{y}{x}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\]

Reproduce

herbie shell --seed 2019163 +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))))