Average Error: 5.2 → 0.1
Time: 17.9s
Precision: 64
\[\frac{x}{y \cdot y} - 3\]
↓
\[\frac{\frac{x}{y}}{y} - 3\]
\frac{x}{y \cdot y} - 3
↓
\frac{\frac{x}{y}}{y} - 3
double f(double x, double y) {
        double r280517 = x;
        double r280518 = y;
        double r280519 = r280518 * r280518;
        double r280520 = r280517 / r280519;
        double r280521 = 3.0;
        double r280522 = r280520 - r280521;
        return r280522;
}
↓
double f(double x, double y) {
        double r280523 = x;
        double r280524 = y;
        double r280525 = r280523 / r280524;
        double r280526 = r280525 / r280524;
        double r280527 = 3.0;
        double r280528 = r280526 - r280527;
        return r280528;
}

Error

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Bits error versus y

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Results

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Target

Original5.2
Target0.1
Herbie0.1
\[\frac{\frac{x}{y}}{y} - 3\]

Derivation

  1. Initial program 5.2

    \[\frac{x}{y \cdot y} - 3\]
  2. Using strategy rm

  3. Applied add-cube-cbrt5.2

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

  4. Applied add-cube-cbrt5.5

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

  5. Applied prod-diff5.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\frac{x}{y \cdot y}} \cdot \sqrt[3]{\frac{x}{y \cdot y}}, \sqrt[3]{\frac{x}{y \cdot y}}, -\sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, \sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right)}\]

  6. Simplified0.1

    \[\leadsto \color{blue}{\left(\frac{\frac{x}{y}}{y} - 3\right)} + \mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, \sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right)\]

  7. Simplified0.1

    \[\leadsto \left(\frac{\frac{x}{y}}{y} - 3\right) + \color{blue}{0}\]

  8. Final simplification0.1

    \[\leadsto \frac{\frac{x}{y}}{y} - 3\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y)
  :name "Statistics.Sample:$skurtosis from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (/ (/ x y) y) 3)

  (- (/ x (* y y)) 3))