Average Error: 33.4 → 0.6
Time: 22.9s
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 r30328218 = x;
        double r30328219 = r30328218 * r30328218;
        double r30328220 = y;
        double r30328221 = r30328220 * r30328220;
        double r30328222 = r30328219 / r30328221;
        double r30328223 = z;
        double r30328224 = r30328223 * r30328223;
        double r30328225 = t;
        double r30328226 = r30328225 * r30328225;
        double r30328227 = r30328224 / r30328226;
        double r30328228 = r30328222 + r30328227;
        return r30328228;
}
↓
double f(double x, double y, double z, double t) {
        double r30328229 = z;
        double r30328230 = t;
        double r30328231 = r30328229 / r30328230;
        double r30328232 = r30328231 * r30328231;
        double r30328233 = x;
        double r30328234 = y;
        double r30328235 = r30328233 / r30328234;
        double r30328236 = r30328235 * r30328235;
        double r30328237 = cbrt(r30328236);
        double r30328238 = cbrt(r30328235);
        double r30328239 = r30328238 * r30328235;
        double r30328240 = r30328237 * r30328239;
        double r30328241 = r30328232 + r30328240;
        return r30328241;
}

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))))