Average Error: 34.0 → 0.7
Time: 5.7s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(\frac{x}{y} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)\right) \cdot \sqrt[3]{\frac{x}{y}}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(\frac{x}{y} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \sqrt[3]{\frac{x}{y}}\right)\right) \cdot \sqrt[3]{\frac{x}{y}}\right)
double f(double x, double y, double z, double t) {
        double r596712 = x;
        double r596713 = r596712 * r596712;
        double r596714 = y;
        double r596715 = r596714 * r596714;
        double r596716 = r596713 / r596715;
        double r596717 = z;
        double r596718 = r596717 * r596717;
        double r596719 = t;
        double r596720 = r596719 * r596719;
        double r596721 = r596718 / r596720;
        double r596722 = r596716 + r596721;
        return r596722;
}
↓
double f(double x, double y, double z, double t) {
        double r596723 = z;
        double r596724 = t;
        double r596725 = r596723 / r596724;
        double r596726 = x;
        double r596727 = y;
        double r596728 = r596726 / r596727;
        double r596729 = cbrt(r596726);
        double r596730 = cbrt(r596727);
        double r596731 = r596729 / r596730;
        double r596732 = cbrt(r596728);
        double r596733 = r596731 * r596732;
        double r596734 = r596728 * r596733;
        double r596735 = r596734 * r596732;
        double r596736 = fma(r596725, r596725, r596735);
        return r596736;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original34.0
Target0.4
Herbie0.7
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 34.0

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

  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

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

  7. Applied associate-*r*0.8

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

  9. Applied cbrt-div0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2020081 +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))))