Average Error: 5.0 → 0.1
Time: 2.2s
Precision: binary64
\[\frac{x}{y \cdot y} - 3\]
↓
\[\frac{1}{y \cdot \frac{y}{x}} - 3\]
\frac{x}{y \cdot y} - 3
↓
\frac{1}{y \cdot \frac{y}{x}} - 3
(FPCore (x y) :precision binary64 (- (/ x (* y y)) 3.0))
↓
(FPCore (x y) :precision binary64 (- (/ 1.0 (* y (/ y x))) 3.0))
double code(double x, double y) {
	return ((double) ((x / ((double) (y * y))) - 3.0));
}
↓
double code(double x, double y) {
	return ((double) ((1.0 / ((double) (y * (y / x)))) - 3.0));
}

Error

Bits error versus x

Bits error versus y

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Results

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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 clear-num_binary645.0

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

  4. Simplified0.1

    \[\leadsto \frac{1}{\color{blue}{\frac{y}{\frac{x}{y}}}} - 3\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt_binary6432.9

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

  7. Applied *-un-lft-identity_binary6432.9

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

  8. Applied times-frac_binary6432.9

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

  9. Applied add-sqr-sqrt_binary6432.9

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

  10. Applied times-frac_binary6432.9

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

  11. Simplified32.9

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020210 
(FPCore (x y)
  :name "Statistics.Sample:$skurtosis from math-functions-0.1.5.2"
  :precision binary64

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

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