Average Error: 19.1 → 3.5
Time: 24.1s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;A \le 4.05646899249745 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left(\left(\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}} \cdot \left(\sqrt{\frac{\frac{\sqrt{A}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt{A}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)\right) \cdot c0\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;A \le 4.05646899249745 \cdot 10^{-310}:\\
\;\;\;\;c0 \cdot \left(\left(\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}} \cdot \left(\sqrt{\frac{\frac{\sqrt{A}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt{A}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)\right) \cdot c0\\

\end{array}
double f(double c0, double A, double V, double l) {
        double r2675697 = c0;
        double r2675698 = A;
        double r2675699 = V;
        double r2675700 = l;
        double r2675701 = r2675699 * r2675700;
        double r2675702 = r2675698 / r2675701;
        double r2675703 = sqrt(r2675702);
        double r2675704 = r2675697 * r2675703;
        return r2675704;
}
↓
double f(double c0, double A, double V, double l) {
        double r2675705 = A;
        double r2675706 = 4.05646899249745e-310;
        bool r2675707 = r2675705 <= r2675706;
        double r2675708 = c0;
        double r2675709 = cbrt(r2675705);
        double r2675710 = l;
        double r2675711 = cbrt(r2675710);
        double r2675712 = r2675709 / r2675711;
        double r2675713 = r2675712 * r2675712;
        double r2675714 = V;
        double r2675715 = cbrt(r2675714);
        double r2675716 = r2675715 * r2675715;
        double r2675717 = cbrt(r2675716);
        double r2675718 = r2675713 / r2675717;
        double r2675719 = sqrt(r2675718);
        double r2675720 = cbrt(r2675715);
        double r2675721 = r2675709 / r2675720;
        double r2675722 = r2675721 / r2675711;
        double r2675723 = sqrt(r2675722);
        double r2675724 = r2675719 * r2675723;
        double r2675725 = 1.0;
        double r2675726 = r2675725 / r2675716;
        double r2675727 = sqrt(r2675726);
        double r2675728 = r2675724 * r2675727;
        double r2675729 = r2675708 * r2675728;
        double r2675730 = sqrt(r2675705);
        double r2675731 = r2675730 / r2675717;
        double r2675732 = r2675711 * r2675711;
        double r2675733 = r2675731 / r2675732;
        double r2675734 = sqrt(r2675733);
        double r2675735 = r2675730 / r2675720;
        double r2675736 = r2675735 / r2675711;
        double r2675737 = sqrt(r2675736);
        double r2675738 = r2675734 * r2675737;
        double r2675739 = r2675727 * r2675738;
        double r2675740 = r2675739 * r2675708;
        double r2675741 = r2675707 ? r2675729 : r2675740;
        return r2675741;
}

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. Split input into 2 regimes

  2. if A < 4.05646899249745e-310

    1. Initial program 19.0

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

    3. Applied *-un-lft-identity19.0

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

    4. Applied times-frac19.3

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

    6. Applied add-cube-cbrt19.6

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

    7. Applied *-un-lft-identity19.6

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

    8. Applied times-frac19.6

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

    9. Applied associate-*l*19.6

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

    10. Simplified18.3

      \[\leadsto c0 \cdot \sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \color{blue}{\frac{\frac{A}{\sqrt[3]{V}}}{\ell}}}\]
    11. Using strategy rm

    12. Applied sqrt-prod13.5

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

    14. Applied add-cube-cbrt13.7

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

    15. Applied add-cube-cbrt13.7

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

    16. Applied cbrt-prod13.7

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

    17. Applied add-cube-cbrt13.8

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

    18. Applied times-frac13.8

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

    19. Applied times-frac12.1

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

    20. Applied sqrt-prod3.6

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

    21. Simplified4.4

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

    if 4.05646899249745e-310 < A

    1. Initial program 19.3

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

    3. Applied *-un-lft-identity19.3

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

    4. Applied times-frac19.0

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

    6. Applied add-cube-cbrt19.3

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

    7. Applied *-un-lft-identity19.3

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

    8. Applied times-frac19.3

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

    9. Applied associate-*l*19.3

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

    10. Simplified18.2

      \[\leadsto c0 \cdot \sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \color{blue}{\frac{\frac{A}{\sqrt[3]{V}}}{\ell}}}\]
    11. Using strategy rm

    12. Applied sqrt-prod13.4

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

    14. Applied add-cube-cbrt13.6

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

    15. Applied add-cube-cbrt13.6

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

    16. Applied cbrt-prod13.6

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

    17. Applied add-sqr-sqrt13.6

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

    18. Applied times-frac13.6

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

    19. Applied times-frac11.7

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

    20. Applied sqrt-prod2.7

      \[\leadsto c0 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}} \cdot \color{blue}{\left(\sqrt{\frac{\frac{\sqrt{A}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt{A}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;A \le 4.05646899249745 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \left(\left(\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}} \cdot \left(\sqrt{\frac{\frac{\sqrt{A}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt{A}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)\right) \cdot c0\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))