Diagrams.Segment:$catParam from diagrams-lib-1.3.0.3, A

Average Error: 10.4 → 0.3

Time: 12.5s

Precision: 64

Math

\[\left(\left(x \cdot 3.0\right) \cdot x\right) \cdot y\]

↓

\[x \cdot \left(\left(y \cdot 3.0\right) \cdot x\right)\]

C

double f(double x, double y) {
        double r34411387 = x;
        double r34411388 = 3.0;
        double r34411389 = r34411387 * r34411388;
        double r34411390 = r34411389 * r34411387;
        double r34411391 = y;
        double r34411392 = r34411390 * r34411391;
        return r34411392;
}

↓

double f(double x, double y) {
        double r34411393 = x;
        double r34411394 = y;
        double r34411395 = 3.0;
        double r34411396 = r34411394 * r34411395;
        double r34411397 = r34411396 * r34411393;
        double r34411398 = r34411393 * r34411397;
        return r34411398;
}

Error

Bits error versus x

Bits error versus y

Target

Original	10.4
Target	0.2
Herbie	0.3

\[\left(x \cdot 3.0\right) \cdot \left(x \cdot y\right)\]

Derivation

1. Initial program 10.4

   \[\left(\left(x \cdot 3.0\right) \cdot x\right) \cdot y\]
2. Using strategy rm

3. Applied associate-*l* 0.2

   \[\leadsto \color{blue}{\left(x \cdot 3.0\right) \cdot \left(x \cdot y\right)}\]
4. Using strategy rm

5. Applied associate-*l* 0.2

   \[\leadsto \color{blue}{x \cdot \left(3.0 \cdot \left(x \cdot y\right)\right)}\]
6. Using strategy rm

7. Applied add-cube-cbrt 0.2

   \[\leadsto x \cdot \left(\color{blue}{\left(\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}\right)} \cdot \left(x \cdot y\right)\right)\]

8. Applied associate-*l* 0.4

   \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \left(\sqrt[3]{3.0} \cdot \left(x \cdot y\right)\right)\right)}\]
9. Using strategy rm

10. Applied pow1 0.4

    \[\leadsto x \cdot \left(\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \left(\sqrt[3]{3.0} \cdot \left(x \cdot \color{blue}{{y}^{1}}\right)\right)\right)\]

11. Applied pow1 0.4

    \[\leadsto x \cdot \left(\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \left(\sqrt[3]{3.0} \cdot \left(\color{blue}{{x}^{1}} \cdot {y}^{1}\right)\right)\right)\]

12. Applied pow-prod-down 0.4

    \[\leadsto x \cdot \left(\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \left(\sqrt[3]{3.0} \cdot \color{blue}{{\left(x \cdot y\right)}^{1}}\right)\right)\]

13. Applied pow1 0.4

    \[\leadsto x \cdot \left(\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{3.0}\right)}^{1}} \cdot {\left(x \cdot y\right)}^{1}\right)\right)\]

14. Applied pow-prod-down 0.4

    \[\leadsto x \cdot \left(\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \color{blue}{{\left(\sqrt[3]{3.0} \cdot \left(x \cdot y\right)\right)}^{1}}\right)\]

15. Applied pow1 0.4

    \[\leadsto x \cdot \left(\left(\sqrt[3]{3.0} \cdot \color{blue}{{\left(\sqrt[3]{3.0}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{3.0} \cdot \left(x \cdot y\right)\right)}^{1}\right)\]

16. Applied pow1 0.4

    \[\leadsto x \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{3.0}\right)}^{1}} \cdot {\left(\sqrt[3]{3.0}\right)}^{1}\right) \cdot {\left(\sqrt[3]{3.0} \cdot \left(x \cdot y\right)\right)}^{1}\right)\]

17. Applied pow-prod-down 0.4

    \[\leadsto x \cdot \left(\color{blue}{{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right)}^{1}} \cdot {\left(\sqrt[3]{3.0} \cdot \left(x \cdot y\right)\right)}^{1}\right)\]

18. Applied pow-prod-down 0.4

    \[\leadsto x \cdot \color{blue}{{\left(\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \left(\sqrt[3]{3.0} \cdot \left(x \cdot y\right)\right)\right)}^{1}}\]

19. Simplified 0.3

    \[\leadsto x \cdot {\color{blue}{\left(\left(3.0 \cdot y\right) \cdot x\right)}}^{1}\]

20. Final simplification 0.3

    \[\leadsto x \cdot \left(\left(y \cdot 3.0\right) \cdot x\right)\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Segment:$catParam from diagrams-lib-1.3.0.3, A"

  :herbie-target
  (* (* x 3.0) (* x y))

  (* (* (* x 3.0) x) y))