Average Error: 33.1 → 0.6
Time: 20.0s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\frac{x}{y} \cdot \frac{x}{y} + \left(\sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}} \cdot \frac{z}{t}\right) \cdot \sqrt[3]{\frac{z}{t}}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\frac{x}{y} \cdot \frac{x}{y} + \left(\sqrt[3]{\frac{z}{t} \cdot \frac{z}{t}} \cdot \frac{z}{t}\right) \cdot \sqrt[3]{\frac{z}{t}}
double f(double x, double y, double z, double t) {
        double r22515120 = x;
        double r22515121 = r22515120 * r22515120;
        double r22515122 = y;
        double r22515123 = r22515122 * r22515122;
        double r22515124 = r22515121 / r22515123;
        double r22515125 = z;
        double r22515126 = r22515125 * r22515125;
        double r22515127 = t;
        double r22515128 = r22515127 * r22515127;
        double r22515129 = r22515126 / r22515128;
        double r22515130 = r22515124 + r22515129;
        return r22515130;
}
↓
double f(double x, double y, double z, double t) {
        double r22515131 = x;
        double r22515132 = y;
        double r22515133 = r22515131 / r22515132;
        double r22515134 = r22515133 * r22515133;
        double r22515135 = z;
        double r22515136 = t;
        double r22515137 = r22515135 / r22515136;
        double r22515138 = r22515137 * r22515137;
        double r22515139 = cbrt(r22515138);
        double r22515140 = r22515139 * r22515137;
        double r22515141 = cbrt(r22515137);
        double r22515142 = r22515140 * r22515141;
        double r22515143 = r22515134 + r22515142;
        return r22515143;
}

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

Original33.1
Target0.4
Herbie0.6
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.1

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

  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

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

  7. Applied associate-*r*0.8

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

  9. Applied cbrt-unprod0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2019170 +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))))