Average Error: 5.0 → 0.1
Time: 11.5s
Precision: 64
\[\frac{x}{y \cdot y} - 3\]
↓
\[\mathsf{fma}\left(3, -1, 3\right) + \left(\frac{1}{\frac{y}{x} \cdot y} - 3\right)\]
\frac{x}{y \cdot y} - 3
↓
\mathsf{fma}\left(3, -1, 3\right) + \left(\frac{1}{\frac{y}{x} \cdot y} - 3\right)
double f(double x, double y) {
        double r219064 = x;
        double r219065 = y;
        double r219066 = r219065 * r219065;
        double r219067 = r219064 / r219066;
        double r219068 = 3.0;
        double r219069 = r219067 - r219068;
        return r219069;
}
↓
double f(double x, double y) {
        double r219070 = 3.0;
        double r219071 = -1.0;
        double r219072 = fma(r219070, r219071, r219070);
        double r219073 = 1.0;
        double r219074 = y;
        double r219075 = x;
        double r219076 = r219074 / r219075;
        double r219077 = r219076 * r219074;
        double r219078 = r219073 / r219077;
        double r219079 = r219078 - r219070;
        double r219080 = r219072 + r219079;
        return r219080;
}

Error

Bits error versus x

Bits error versus y

Target

Original5.0
Target0.1
Herbie0.1
\[\frac{\frac{x}{y}}{y} - 3\]

Derivation

  1. Initial program 5.0

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

  3. Applied add-cube-cbrt5.0

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

  4. Applied add-sqr-sqrt21.6

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

  5. Applied prod-diff21.6

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

  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

    \[\leadsto \left(\frac{\frac{x}{y}}{y} - 3\right) + \color{blue}{\mathsf{fma}\left(3, -1, 3\right)}\]
  8. Using strategy rm

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

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

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

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

  11. Applied times-frac0.1

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

  12. Applied associate-/l*0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y)
  :name "Statistics.Sample:$skurtosis from math-functions-0.1.5.2"

  :herbie-target
  (- (/ (/ x y) y) 3.0)

  (- (/ x (* y y)) 3.0))