Average Error: 4.9 → 0.1
Time: 1.6m
Precision: 64
\[\frac{x}{y \cdot y} - 3\]
↓
\[\frac{\frac{1}{y}}{\frac{y}{x}} - 3\]
\frac{x}{y \cdot y} - 3
↓
\frac{\frac{1}{y}}{\frac{y}{x}} - 3
double f(double x, double y) {
        double r500078 = x;
        double r500079 = y;
        double r500080 = r500079 * r500079;
        double r500081 = r500078 / r500080;
        double r500082 = 3.0;
        double r500083 = r500081 - r500082;
        return r500083;
}
↓
double f(double x, double y) {
        double r500084 = 1.0;
        double r500085 = y;
        double r500086 = r500084 / r500085;
        double r500087 = x;
        double r500088 = r500085 / r500087;
        double r500089 = r500086 / r500088;
        double r500090 = 3.0;
        double r500091 = r500089 - r500090;
        return r500091;
}

Error

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Results

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Target

Original4.9
Target0.1
Herbie0.1
\[\frac{\frac{x}{y}}{y} - 3\]

Derivation

  1. Initial program 4.9

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

  3. Applied *-un-lft-identity4.9

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

  4. Applied *-un-lft-identity4.9

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

  5. Applied distribute-lft-out--4.9

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

  6. Simplified0.1

    \[\leadsto 1 \cdot \color{blue}{\left(\frac{\frac{x}{y}}{y} - 3\right)}\]
  7. Using strategy rm

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

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

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

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

  10. Applied times-frac0.1

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

  11. Applied associate-/l*0.1

    \[\leadsto 1 \cdot \left(\color{blue}{\frac{\frac{1}{1}}{\frac{y}{\frac{x}{y}}}} - 3\right)\]
  12. Using strategy rm

  13. Applied div-inv0.1

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

  14. Applied associate-/r*0.1

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

  15. Taylor expanded around 0 0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019209 +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))