Average Error: 10.8 → 0.2
Time: 2.1s
Precision: 64
\[\frac{x}{y \cdot y}\]
↓
\[\frac{\frac{x}{y}}{y}\]
\frac{x}{y \cdot y}
↓
\frac{\frac{x}{y}}{y}
double f(double x, double y) {
        double r330827 = x;
        double r330828 = y;
        double r330829 = r330828 * r330828;
        double r330830 = r330827 / r330829;
        return r330830;
}
↓
double f(double x, double y) {
        double r330831 = x;
        double r330832 = y;
        double r330833 = r330831 / r330832;
        double r330834 = r330833 / r330832;
        return r330834;
}

Error

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Bits error versus y

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Results

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Target

Original10.8
Target0.2
Herbie0.2
\[\frac{\frac{x}{y}}{y}\]

Derivation

  1. Initial program 10.8

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

  3. Applied associate-/r*0.2

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

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

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

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

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

  7. Applied times-frac0.2

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

  8. Applied associate-/l*0.6

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

  10. Applied div-inv0.7

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

  11. Applied add-sqr-sqrt0.7

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

  12. Applied times-frac0.3

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

  13. Simplified0.3

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

  14. Simplified0.2

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

  16. Applied associate-*r/0.2

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

  17. Simplified0.2

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

  18. Final simplification0.2

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

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x y)
  :name "Physics.ForceLayout:coulombForce from force-layout-0.4.0.2"
  :precision binary64

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

  (/ x (* y y)))