Average Error: 4.8 → 0.1
Time: 32.5s
Precision: 64
\[\frac{x}{y \cdot y} - 3\]
↓
\[\mathsf{fma}\left(1, \frac{1}{\frac{y}{\frac{x}{y}}}, -{\left(\sqrt[3]{3}\right)}^{3}\right) + \mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, {\left(\sqrt[3]{3}\right)}^{3}\right)\]
\frac{x}{y \cdot y} - 3
↓
\mathsf{fma}\left(1, \frac{1}{\frac{y}{\frac{x}{y}}}, -{\left(\sqrt[3]{3}\right)}^{3}\right) + \mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, {\left(\sqrt[3]{3}\right)}^{3}\right)
double f(double x, double y) {
        double r193675 = x;
        double r193676 = y;
        double r193677 = r193676 * r193676;
        double r193678 = r193675 / r193677;
        double r193679 = 3.0;
        double r193680 = r193678 - r193679;
        return r193680;
}
↓
double f(double x, double y) {
        double r193681 = 1.0;
        double r193682 = y;
        double r193683 = x;
        double r193684 = r193683 / r193682;
        double r193685 = r193682 / r193684;
        double r193686 = r193681 / r193685;
        double r193687 = 3.0;
        double r193688 = cbrt(r193687);
        double r193689 = 3.0;
        double r193690 = pow(r193688, r193689);
        double r193691 = -r193690;
        double r193692 = fma(r193681, r193686, r193691);
        double r193693 = -r193688;
        double r193694 = r193688 * r193688;
        double r193695 = fma(r193693, r193694, r193690);
        double r193696 = r193692 + r193695;
        return r193696;
}

Error

Bits error versus x

Bits error versus y

Target

Original4.8
Target0.1
Herbie0.1
\[\frac{\frac{x}{y}}{y} - 3\]

Derivation

  1. Initial program 4.8

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

  3. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} - 3\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.1

    \[\leadsto \frac{\frac{x}{\color{blue}{1 \cdot y}}}{y} - 3\]

  6. Applied *-un-lft-identity0.1

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot x}}{1 \cdot y}}{y} - 3\]

  7. Applied times-frac0.1

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{x}{y}}}{y} - 3\]

  8. Applied associate-/l*0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{y}{\frac{x}{y}}}} - 3\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.1

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

  11. Applied *-un-lft-identity0.1

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

  12. Applied prod-diff0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{\frac{1}{1}}{\frac{y}{\frac{x}{y}}}, -\sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, \sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right)}\]

  13. Simplified0.8

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{\frac{1}{1}}{\frac{y}{\frac{x}{y}}}, -{\left(\sqrt[3]{3}\right)}^{3}\right)} + \mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, \sqrt[3]{3} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right)\]

  14. Simplified0.1

    \[\leadsto \mathsf{fma}\left(1, \frac{\frac{1}{1}}{\frac{y}{\frac{x}{y}}}, -{\left(\sqrt[3]{3}\right)}^{3}\right) + \color{blue}{\mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, {\left(\sqrt[3]{3}\right)}^{3}\right)}\]

  15. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(1, \frac{1}{\frac{y}{\frac{x}{y}}}, -{\left(\sqrt[3]{3}\right)}^{3}\right) + \mathsf{fma}\left(-\sqrt[3]{3}, \sqrt[3]{3} \cdot \sqrt[3]{3}, {\left(\sqrt[3]{3}\right)}^{3}\right)\]

Reproduce

herbie shell --seed 2019322 +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))