Average Error: 18.9 → 2.3
Time: 1.4m
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[c0 \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}} \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
c0 \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}} \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}
double f(double c0, double A, double V, double l) {
        double r15430677 = c0;
        double r15430678 = A;
        double r15430679 = V;
        double r15430680 = l;
        double r15430681 = r15430679 * r15430680;
        double r15430682 = r15430678 / r15430681;
        double r15430683 = sqrt(r15430682);
        double r15430684 = r15430677 * r15430683;
        return r15430684;
}
↓
double f(double c0, double A, double V, double l) {
        double r15430685 = c0;
        double r15430686 = A;
        double r15430687 = cbrt(r15430686);
        double r15430688 = V;
        double r15430689 = cbrt(r15430688);
        double r15430690 = r15430687 / r15430689;
        double r15430691 = l;
        double r15430692 = cbrt(r15430691);
        double r15430693 = r15430690 / r15430692;
        double r15430694 = sqrt(r15430693);
        double r15430695 = fabs(r15430690);
        double r15430696 = r15430694 * r15430695;
        double r15430697 = r15430692 * r15430692;
        double r15430698 = sqrt(r15430697);
        double r15430699 = r15430696 / r15430698;
        double r15430700 = r15430685 * r15430699;
        return r15430700;
}

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 18.9

    \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity18.9

    \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}\]

  4. Applied times-frac18.8

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt19.1

    \[\leadsto c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\]

  7. Applied *-un-lft-identity19.1

    \[\leadsto c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{\color{blue}{1 \cdot A}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

  8. Applied times-frac19.1

    \[\leadsto c0 \cdot \sqrt{\frac{1}{V} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{A}{\sqrt[3]{\ell}}\right)}}\]

  9. Applied associate-*r*17.8

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\frac{1}{V} \cdot \frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{A}{\sqrt[3]{\ell}}}}\]
  10. Using strategy rm

  11. Applied un-div-inv17.8

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{A}{\sqrt[3]{\ell}}}\]

  12. Applied associate-*l/17.7

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V} \cdot \frac{A}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\]

  13. Applied sqrt-div13.0

    \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\frac{1}{V} \cdot \frac{A}{\sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\]

  14. Simplified13.5

    \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]
  15. Using strategy rm

  16. Applied *-un-lft-identity13.5

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\color{blue}{1 \cdot \ell}}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

  17. Applied cbrt-prod13.5

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{A}{V}}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{\ell}}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

  18. Applied add-cube-cbrt13.6

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{A}{\color{blue}{\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \sqrt[3]{V}}}}{\sqrt[3]{1} \cdot \sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

  19. Applied add-cube-cbrt13.7

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{\color{blue}{\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A}}}{\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \sqrt[3]{V}}}{\sqrt[3]{1} \cdot \sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

  20. Applied times-frac13.7

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V}}}}{\sqrt[3]{1} \cdot \sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

  21. Applied times-frac11.5

    \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{1}} \cdot \frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

  22. Applied sqrt-prod3.9

    \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{1}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

  23. Simplified2.3

    \[\leadsto c0 \cdot \frac{\color{blue}{\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

  24. Final simplification2.3

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}} \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))