Average Error: 4.9 → 0.1
Time: 2.3s
Precision: 64
\[\frac{x}{y \cdot y} - 3\]
↓
\[\frac{1}{y \cdot \frac{y}{x}} - 3\]
\frac{x}{y \cdot y} - 3
↓
\frac{1}{y \cdot \frac{y}{x}} - 3
double f(double x, double y) {
        double r282404 = x;
        double r282405 = y;
        double r282406 = r282405 * r282405;
        double r282407 = r282404 / r282406;
        double r282408 = 3.0;
        double r282409 = r282407 - r282408;
        return r282409;
}
↓
double f(double x, double y) {
        double r282410 = 1.0;
        double r282411 = y;
        double r282412 = x;
        double r282413 = r282411 / r282412;
        double r282414 = r282411 * r282413;
        double r282415 = r282410 / r282414;
        double r282416 = 3.0;
        double r282417 = r282415 - r282416;
        return r282417;
}

Error

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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 clear-num0.1

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

  7. Applied add-sqr-sqrt32.3

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

  8. Applied *-un-lft-identity32.3

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

  9. Applied times-frac32.3

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

  10. Applied add-sqr-sqrt32.3

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

  11. Applied times-frac32.3

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

  12. Simplified32.3

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020034 
(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))