Average Error: 0.3 → 0.2
Time: 1.4s
Precision: 64
\[\frac{x}{y \cdot 3}\]
↓
\[\frac{\frac{x}{3}}{y}\]
\frac{x}{y \cdot 3}
↓
\frac{\frac{x}{3}}{y}
double f(double x, double y) {
        double r750100 = x;
        double r750101 = y;
        double r750102 = 3.0;
        double r750103 = r750101 * r750102;
        double r750104 = r750100 / r750103;
        return r750104;
}
↓
double f(double x, double y) {
        double r750105 = x;
        double r750106 = 3.0;
        double r750107 = r750105 / r750106;
        double r750108 = y;
        double r750109 = r750107 / r750108;
        return r750109;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.3
Target0.2
Herbie0.2
\[\frac{\frac{x}{y}}{3}\]

Derivation

  1. Initial program 0.3

    \[\frac{x}{y \cdot 3}\]
  2. Using strategy rm

  3. Applied associate-/r*0.2

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

  5. Applied *-un-lft-identity0.2

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

  6. Applied div-inv0.3

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

  7. Applied times-frac0.3

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

  8. Simplified0.3

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

  10. Applied div-inv0.4

    \[\leadsto x \cdot \color{blue}{\left(\frac{1}{y} \cdot \frac{1}{3}\right)}\]

  11. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y}\right) \cdot \frac{1}{3}}\]

  12. Simplified0.3

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

  14. Applied associate-*l/0.3

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020064 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, C"
  :precision binary64

  :herbie-target
  (/ (/ x y) 3)

  (/ x (* y 3)))