Average Error: 5.1 → 0.1
Time: 2.8s
Precision: 64
\[\frac{x}{y \cdot y} - 3\]
↓
\[\frac{\frac{1}{\frac{y}{x}}}{y} - 3\]
\frac{x}{y \cdot y} - 3
↓
\frac{\frac{1}{\frac{y}{x}}}{y} - 3
double code(double x, double y) {
	return ((x / (y * y)) - 3.0);
}
↓
double code(double x, double y) {
	return (((1.0 / (y / x)) / y) - 3.0);
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

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

Derivation

  1. Initial program 5.1

    \[\frac{x}{y \cdot y} - 3\]
  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 associate-/r/0.1

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

  11. Applied associate-/r*0.1

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020060 
(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))