Average Error: 33.4 → 0.6
Time: 24.0s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\frac{z}{t} \cdot \frac{z}{t} + \sqrt[3]{\frac{x}{y} \cdot \frac{x}{y}} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \frac{x}{y}\right)\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\frac{z}{t} \cdot \frac{z}{t} + \sqrt[3]{\frac{x}{y} \cdot \frac{x}{y}} \cdot \left(\sqrt[3]{\frac{x}{y}} \cdot \frac{x}{y}\right)
double f(double x, double y, double z, double t) {
        double r32606957 = x;
        double r32606958 = r32606957 * r32606957;
        double r32606959 = y;
        double r32606960 = r32606959 * r32606959;
        double r32606961 = r32606958 / r32606960;
        double r32606962 = z;
        double r32606963 = r32606962 * r32606962;
        double r32606964 = t;
        double r32606965 = r32606964 * r32606964;
        double r32606966 = r32606963 / r32606965;
        double r32606967 = r32606961 + r32606966;
        return r32606967;
}
↓
double f(double x, double y, double z, double t) {
        double r32606968 = z;
        double r32606969 = t;
        double r32606970 = r32606968 / r32606969;
        double r32606971 = r32606970 * r32606970;
        double r32606972 = x;
        double r32606973 = y;
        double r32606974 = r32606972 / r32606973;
        double r32606975 = r32606974 * r32606974;
        double r32606976 = cbrt(r32606975);
        double r32606977 = cbrt(r32606974);
        double r32606978 = r32606977 * r32606974;
        double r32606979 = r32606976 * r32606978;
        double r32606980 = r32606971 + r32606979;
        return r32606980;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.4
Target0.4
Herbie0.6
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.4

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

  2. 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 fma-udef0.4

    \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t} + \frac{x}{y} \cdot \frac{x}{y}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.8

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

  7. Applied associate-*l*0.8

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

  9. Applied cbrt-unprod0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 +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) 2.0) (pow (/ z t) 2.0))

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