Average Error: 32.2 → 0.9
Time: 24.3s
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 r33042919 = x;
        double r33042920 = r33042919 * r33042919;
        double r33042921 = y;
        double r33042922 = r33042921 * r33042921;
        double r33042923 = r33042920 / r33042922;
        double r33042924 = z;
        double r33042925 = r33042924 * r33042924;
        double r33042926 = t;
        double r33042927 = r33042926 * r33042926;
        double r33042928 = r33042925 / r33042927;
        double r33042929 = r33042923 + r33042928;
        return r33042929;
}
↓
double f(double x, double y, double z, double t) {
        double r33042930 = z;
        double r33042931 = t;
        double r33042932 = r33042930 / r33042931;
        double r33042933 = x;
        double r33042934 = cbrt(r33042933);
        double r33042935 = y;
        double r33042936 = cbrt(r33042935);
        double r33042937 = r33042934 / r33042936;
        double r33042938 = r33042935 / r33042933;
        double r33042939 = r33042937 / r33042938;
        double r33042940 = r33042939 * r33042937;
        double r33042941 = r33042937 * r33042940;
        double r33042942 = fma(r33042932, r33042932, r33042941);
        return r33042942;
}

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))))