Average Error: 33.4 → 1.5
Time: 5.1s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\frac{x}{y}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}}\right)\]
\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 r623762 = x;
        double r623763 = r623762 * r623762;
        double r623764 = y;
        double r623765 = r623764 * r623764;
        double r623766 = r623763 / r623765;
        double r623767 = z;
        double r623768 = r623767 * r623767;
        double r623769 = t;
        double r623770 = r623769 * r623769;
        double r623771 = r623768 / r623770;
        double r623772 = r623766 + r623771;
        return r623772;
}
↓
double f(double x, double y, double z, double t) {
        double r623773 = z;
        double r623774 = t;
        double r623775 = r623773 / r623774;
        double r623776 = x;
        double r623777 = y;
        double r623778 = r623776 / r623777;
        double r623779 = cbrt(r623777);
        double r623780 = r623779 * r623779;
        double r623781 = r623778 / r623780;
        double r623782 = r623776 / r623779;
        double r623783 = r623781 * r623782;
        double r623784 = fma(r623775, r623775, r623783);
        return r623784;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original33.4
Target0.4
Herbie1.5
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.4

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

  2. Simplified19.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)}\]
  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. Simplified1.5

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

  11. Final simplification1.5

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

Reproduce

herbie shell --seed 2020065 +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))))