Average Error: 34.3 → 1.4
Time: 6.6s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}}\right)
double f(double x, double y, double z, double t) {
        double r651863 = x;
        double r651864 = r651863 * r651863;
        double r651865 = y;
        double r651866 = r651865 * r651865;
        double r651867 = r651864 / r651866;
        double r651868 = z;
        double r651869 = r651868 * r651868;
        double r651870 = t;
        double r651871 = r651870 * r651870;
        double r651872 = r651869 / r651871;
        double r651873 = r651867 + r651872;
        return r651873;
}
↓
double f(double x, double y, double z, double t) {
        double r651874 = z;
        double r651875 = t;
        double r651876 = r651874 / r651875;
        double r651877 = x;
        double r651878 = y;
        double r651879 = r651877 / r651878;
        double r651880 = cbrt(r651878);
        double r651881 = r651880 * r651880;
        double r651882 = r651879 / r651881;
        double r651883 = r651877 / r651880;
        double r651884 = r651882 * r651883;
        double r651885 = fma(r651876, r651876, r651884);
        return r651885;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original34.3
Target0.4
Herbie1.4
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 34.3

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

  2. Simplified19.5

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

  4. Applied times-frac0.4

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

  6. Applied add-cube-cbrt0.8

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

  7. Applied *-un-lft-identity0.8

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

  8. Applied times-frac0.8

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

  9. Applied associate-*r*1.5

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

  10. Simplified1.4

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

  11. Final simplification1.4

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

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

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