Average Error: 10.1 → 0.2
Time: 1.9s
Precision: 64
\[\frac{x}{y \cdot y}\]
↓
\[\frac{1}{y} \cdot \frac{x}{y}\]
\frac{x}{y \cdot y}
↓
\frac{1}{y} \cdot \frac{x}{y}
double f(double x, double y) {
        double r280937 = x;
        double r280938 = y;
        double r280939 = r280938 * r280938;
        double r280940 = r280937 / r280939;
        return r280940;
}
↓
double f(double x, double y) {
        double r280941 = 1.0;
        double r280942 = y;
        double r280943 = r280941 / r280942;
        double r280944 = x;
        double r280945 = r280944 / r280942;
        double r280946 = r280943 * r280945;
        return r280946;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original10.1
Target0.2
Herbie0.2
\[\frac{\frac{x}{y}}{y}\]

Derivation

  1. Initial program 10.1

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

  3. Applied associate-/r*0.2

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

  5. Applied add-sqr-sqrt32.8

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

  6. Applied add-sqr-sqrt32.9

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

  7. Applied *-un-lft-identity32.9

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

  8. Applied times-frac32.9

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

  9. Applied times-frac32.9

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

  10. Simplified32.8

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019356 +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)))