Average Error: 4.8 → 0.1
Time: 3.3s
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 r244043 = x;
        double r244044 = y;
        double r244045 = r244044 * r244044;
        double r244046 = r244043 / r244045;
        double r244047 = 3.0;
        double r244048 = r244046 - r244047;
        return r244048;
}
↓
double f(double x, double y) {
        double r244049 = 1.0;
        double r244050 = y;
        double r244051 = r244049 / r244050;
        double r244052 = x;
        double r244053 = r244050 / r244052;
        double r244054 = r244051 / r244053;
        double r244055 = 3.0;
        double r244056 = r244054 - r244055;
        return r244056;
}

Error

Bits error versus x

Bits error versus y

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Results

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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 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 2020021 +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))