Average Error: 4.9 → 0.1
Time: 15.1s
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 r226049 = x;
        double r226050 = y;
        double r226051 = r226050 * r226050;
        double r226052 = r226049 / r226051;
        double r226053 = 3.0;
        double r226054 = r226052 - r226053;
        return r226054;
}
↓
double f(double x, double y) {
        double r226055 = 1.0;
        double r226056 = y;
        double r226057 = r226055 / r226056;
        double r226058 = x;
        double r226059 = r226056 / r226058;
        double r226060 = r226057 / r226059;
        double r226061 = 3.0;
        double r226062 = r226060 - r226061;
        return r226062;
}

Error

Bits error versus x

Bits error versus y

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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 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 div-inv0.1

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

  11. Applied associate-/r*0.1

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

  12. Simplified0.1

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

  13. Taylor expanded around 0 0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019209 
(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))