Average Error: 33.8 → 2.0
Time: 19.2s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\frac{\sqrt[3]{z}}{\frac{\sqrt[3]{t}}{z}}}{t}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\frac{\sqrt[3]{z}}{\frac{\sqrt[3]{t}}{z}}}{t}\right)
double f(double x, double y, double z, double t) {
        double r680725 = x;
        double r680726 = r680725 * r680725;
        double r680727 = y;
        double r680728 = r680727 * r680727;
        double r680729 = r680726 / r680728;
        double r680730 = z;
        double r680731 = r680730 * r680730;
        double r680732 = t;
        double r680733 = r680732 * r680732;
        double r680734 = r680731 / r680733;
        double r680735 = r680729 + r680734;
        return r680735;
}
↓
double f(double x, double y, double z, double t) {
        double r680736 = x;
        double r680737 = y;
        double r680738 = r680736 / r680737;
        double r680739 = z;
        double r680740 = cbrt(r680739);
        double r680741 = r680740 * r680740;
        double r680742 = t;
        double r680743 = cbrt(r680742);
        double r680744 = r680743 * r680743;
        double r680745 = r680741 / r680744;
        double r680746 = r680743 / r680739;
        double r680747 = r680740 / r680746;
        double r680748 = r680747 / r680742;
        double r680749 = r680745 * r680748;
        double r680750 = fma(r680738, r680738, r680749);
        return r680750;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.8
Target0.4
Herbie2.0
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.8

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

  2. Simplified19.2

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

  4. Applied associate-/r*13.5

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

  5. Simplified4.3

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

  7. Applied *-un-lft-identity4.3

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

  8. Applied *-un-lft-identity4.3

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

  9. Applied add-cube-cbrt4.7

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

  10. Applied times-frac4.7

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

  11. Applied add-cube-cbrt4.8

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

  12. Applied times-frac4.8

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

  13. Applied times-frac2.0

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

  14. Simplified2.0

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

  15. Final simplification2.0

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

Reproduce

herbie shell --seed 2020045 +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))))