Average Error: 33.9 → 1.5
Time: 18.8s
Precision: 64
\[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\]
↓
\[\frac{x}{y} \cdot \frac{x}{y} + \frac{\frac{z}{t}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\]
\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}
↓
\frac{x}{y} \cdot \frac{x}{y} + \frac{\frac{z}{t}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}
double f(double x, double y, double z, double t) {
        double r371286 = x;
        double r371287 = r371286 * r371286;
        double r371288 = y;
        double r371289 = r371288 * r371288;
        double r371290 = r371287 / r371289;
        double r371291 = z;
        double r371292 = r371291 * r371291;
        double r371293 = t;
        double r371294 = r371293 * r371293;
        double r371295 = r371292 / r371294;
        double r371296 = r371290 + r371295;
        return r371296;
}
↓
double f(double x, double y, double z, double t) {
        double r371297 = x;
        double r371298 = y;
        double r371299 = r371297 / r371298;
        double r371300 = r371299 * r371299;
        double r371301 = z;
        double r371302 = t;
        double r371303 = r371301 / r371302;
        double r371304 = cbrt(r371302);
        double r371305 = r371304 * r371304;
        double r371306 = r371303 / r371305;
        double r371307 = r371301 / r371304;
        double r371308 = r371306 * r371307;
        double r371309 = r371300 + r371308;
        return r371309;
}

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

Original33.9
Target0.4
Herbie1.5
\[{\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2}\]

Derivation

  1. Initial program 33.9

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

  2. Simplified13.4

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

  3. Taylor expanded around 0 33.9

    \[\leadsto \color{blue}{\frac{{x}^{2}}{{y}^{2}} + \frac{{z}^{2}}{{t}^{2}}}\]

  4. Simplified0.4

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

  6. Applied add-cube-cbrt0.8

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

  7. Applied *-un-lft-identity0.8

    \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \frac{\color{blue}{1 \cdot z}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\]

  8. Applied times-frac0.8

    \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{z}{t} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\right)}\]

  9. Applied associate-*r*1.5

    \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \color{blue}{\left(\frac{z}{t} \cdot \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{z}{\sqrt[3]{t}}}\]

  10. Simplified1.5

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

  11. Final simplification1.5

    \[\leadsto \frac{x}{y} \cdot \frac{x}{y} + \frac{\frac{z}{t}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{z}{\sqrt[3]{t}}\]

Reproduce

herbie shell --seed 2019196 +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))))