Average Error: 0.3 → 0.3
Time: 7.3s
Precision: 64
\[\left(\left(x \cdot 3\right) \cdot y\right) \cdot y\]
↓
\[y \cdot \left(\left(x \cdot y\right) \cdot 3\right)\]
\left(\left(x \cdot 3\right) \cdot y\right) \cdot y
↓
y \cdot \left(\left(x \cdot y\right) \cdot 3\right)
double f(double x, double y) {
        double r937601 = x;
        double r937602 = 3.0;
        double r937603 = r937601 * r937602;
        double r937604 = y;
        double r937605 = r937603 * r937604;
        double r937606 = r937605 * r937604;
        return r937606;
}
↓
double f(double x, double y) {
        double r937607 = y;
        double r937608 = x;
        double r937609 = r937608 * r937607;
        double r937610 = 3.0;
        double r937611 = r937609 * r937610;
        double r937612 = r937607 * r937611;
        return r937612;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.3
Target0.2
Herbie0.3
\[\left(x \cdot \left(3 \cdot y\right)\right) \cdot y\]

Derivation

  1. Initial program 0.3

    \[\left(\left(x \cdot 3\right) \cdot y\right) \cdot y\]
  2. Using strategy rm

  3. Applied pow10.3

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

  4. Applied pow10.3

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

  5. Applied pow10.3

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

  6. Applied pow10.3

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

  7. Applied pow-prod-down0.3

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

  8. Applied pow-prod-down0.3

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

  9. Applied pow-prod-down0.3

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

  10. Simplified0.2

    \[\leadsto {\color{blue}{\left(\left(y \cdot x\right) \cdot \left(y \cdot 3\right)\right)}}^{1}\]
  11. Using strategy rm

  12. Applied associate-*l*0.2

    \[\leadsto {\color{blue}{\left(y \cdot \left(x \cdot \left(y \cdot 3\right)\right)\right)}}^{1}\]
  13. Using strategy rm

  14. Applied associate-*r*0.3

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

  15. Final simplification0.3

    \[\leadsto y \cdot \left(\left(x \cdot y\right) \cdot 3\right)\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Segment:$catParam from diagrams-lib-1.3.0.3, B"
  :precision binary64

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

  (* (* (* x 3) y) y))