Average Error: 15.3 → 2.8
Time: 3.5s
Precision: 64
\[\frac{\left(x \cdot 2\right) \cdot y}{x - y}\]
↓
\[\left(\left(x \cdot 2\right) \cdot \frac{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{y}}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{x - y}}\]
\frac{\left(x \cdot 2\right) \cdot y}{x - y}
↓
\left(\left(x \cdot 2\right) \cdot \frac{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{y}}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{x - y}}
double f(double x, double y) {
        double r488285 = x;
        double r488286 = 2.0;
        double r488287 = r488285 * r488286;
        double r488288 = y;
        double r488289 = r488287 * r488288;
        double r488290 = r488285 - r488288;
        double r488291 = r488289 / r488290;
        return r488291;
}
↓
double f(double x, double y) {
        double r488292 = x;
        double r488293 = 2.0;
        double r488294 = r488292 * r488293;
        double r488295 = y;
        double r488296 = cbrt(r488295);
        double r488297 = r488296 * r488296;
        double r488298 = cbrt(r488297);
        double r488299 = cbrt(r488296);
        double r488300 = r488298 * r488299;
        double r488301 = r488300 * r488296;
        double r488302 = r488292 - r488295;
        double r488303 = cbrt(r488302);
        double r488304 = r488303 * r488303;
        double r488305 = r488301 / r488304;
        double r488306 = r488294 * r488305;
        double r488307 = r488296 / r488303;
        double r488308 = r488306 * r488307;
        return r488308;
}

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.3
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;x \lt -1.7210442634149447 \cdot 10^{81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x \lt 83645045635564432:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array}\]

Derivation

  1. Initial program 15.3

    \[\frac{\left(x \cdot 2\right) \cdot y}{x - y}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity15.3

    \[\leadsto \frac{\left(x \cdot 2\right) \cdot y}{\color{blue}{1 \cdot \left(x - y\right)}}\]

  4. Applied times-frac7.4

    \[\leadsto \color{blue}{\frac{x \cdot 2}{1} \cdot \frac{y}{x - y}}\]

  5. Simplified7.4

    \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot \frac{y}{x - y}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt8.6

    \[\leadsto \left(x \cdot 2\right) \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \sqrt[3]{x - y}}}\]

  8. Applied add-cube-cbrt8.0

    \[\leadsto \left(x \cdot 2\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \sqrt[3]{x - y}}\]

  9. Applied times-frac8.0

    \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{x - y}}\right)}\]

  10. Applied associate-*r*2.4

    \[\leadsto \color{blue}{\left(\left(x \cdot 2\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{x - y}}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt2.6

    \[\leadsto \left(\left(x \cdot 2\right) \cdot \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} \cdot \sqrt[3]{y}}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{x - y}}\]

  13. Applied cbrt-prod2.8

    \[\leadsto \left(\left(x \cdot 2\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)} \cdot \sqrt[3]{y}}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{x - y}}\]

  14. Final simplification2.8

    \[\leadsto \left(\left(x \cdot 2\right) \cdot \frac{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{y}}{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{x - y}}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2 x) (- x y)) y) (if (< x 83645045635564432) (/ (* x 2) (/ (- x y) y)) (* (/ (* 2 x) (- x y)) y)))

  (/ (* (* x 2) y) (- x y)))