Average Error: 19.1 → 13.9
Time: 4.8s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \le -5.556283518325683243464468114976330835385 \cdot 10^{-309}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \frac{\frac{A}{\ell}}{\sqrt[3]{V}}}}\right) \cdot \sqrt{\sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}}\\ \mathbf{elif}\;V \le 3.621419097460427185242147570218797506351 \cdot 10^{-142}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V}} \cdot \sqrt{\frac{A}{\ell}}\right)\\ \mathbf{elif}\;V \le 2.480087127913832854241189970085787477828 \cdot 10^{154}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V}} \cdot \sqrt{\frac{A}{\ell}}\right)\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \le -5.556283518325683243464468114976330835385 \cdot 10^{-309}:\\
\;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \frac{\frac{A}{\ell}}{\sqrt[3]{V}}}}\right) \cdot \sqrt{\sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}}\\

\mathbf{elif}\;V \le 3.621419097460427185242147570218797506351 \cdot 10^{-142}:\\
\;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V}} \cdot \sqrt{\frac{A}{\ell}}\right)\\

\mathbf{elif}\;V \le 2.480087127913832854241189970085787477828 \cdot 10^{154}:\\
\;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\right)\\

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

\end{array}
double f(double c0, double A, double V, double l) {
        double r257706 = c0;
        double r257707 = A;
        double r257708 = V;
        double r257709 = l;
        double r257710 = r257708 * r257709;
        double r257711 = r257707 / r257710;
        double r257712 = sqrt(r257711);
        double r257713 = r257706 * r257712;
        return r257713;
}
↓
double f(double c0, double A, double V, double l) {
        double r257714 = V;
        double r257715 = -5.556283518325683e-309;
        bool r257716 = r257714 <= r257715;
        double r257717 = c0;
        double r257718 = 1.0;
        double r257719 = cbrt(r257714);
        double r257720 = r257719 * r257719;
        double r257721 = r257718 / r257720;
        double r257722 = A;
        double r257723 = l;
        double r257724 = r257722 / r257723;
        double r257725 = r257724 / r257719;
        double r257726 = r257721 * r257725;
        double r257727 = sqrt(r257726);
        double r257728 = sqrt(r257727);
        double r257729 = r257717 * r257728;
        double r257730 = r257718 / r257714;
        double r257731 = r257730 * r257724;
        double r257732 = sqrt(r257731);
        double r257733 = sqrt(r257732);
        double r257734 = r257729 * r257733;
        double r257735 = 3.621419097460427e-142;
        bool r257736 = r257714 <= r257735;
        double r257737 = sqrt(r257730);
        double r257738 = sqrt(r257724);
        double r257739 = r257737 * r257738;
        double r257740 = r257717 * r257739;
        double r257741 = 2.480087127913833e+154;
        bool r257742 = r257714 <= r257741;
        double r257743 = cbrt(r257722);
        double r257744 = r257743 * r257743;
        double r257745 = r257744 / r257714;
        double r257746 = sqrt(r257745);
        double r257747 = r257743 / r257723;
        double r257748 = sqrt(r257747);
        double r257749 = r257746 * r257748;
        double r257750 = r257717 * r257749;
        double r257751 = r257742 ? r257750 : r257740;
        double r257752 = r257736 ? r257740 : r257751;
        double r257753 = r257716 ? r257734 : r257752;
        return r257753;
}

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 3 regimes

  2. if V < -5.556283518325683e-309

    1. Initial program 19.6

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

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

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

    4. Applied times-frac19.6

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

    6. Applied add-sqr-sqrt19.8

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

    7. Applied associate-*r*19.8

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

    9. Applied add-cube-cbrt19.8

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

    10. Applied *-un-lft-identity19.8

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

    11. Applied times-frac19.8

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

    12. Applied associate-*l*19.8

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

    13. Simplified19.8

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

    if -5.556283518325683e-309 < V < 3.621419097460427e-142 or 2.480087127913833e+154 < V

    1. Initial program 23.6

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

    3. Applied *-un-lft-identity23.6

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

    4. Applied times-frac23.0

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

    5. Applied sqrt-prod11.4

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

    if 3.621419097460427e-142 < V < 2.480087127913833e+154

    1. Initial program 13.3

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

    3. Applied add-cube-cbrt13.7

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

    4. Applied times-frac11.8

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

    5. Applied sqrt-prod4.1

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

  4. Final simplification13.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \le -5.556283518325683243464468114976330835385 \cdot 10^{-309}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \frac{\frac{A}{\ell}}{\sqrt[3]{V}}}}\right) \cdot \sqrt{\sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}}\\ \mathbf{elif}\;V \le 3.621419097460427185242147570218797506351 \cdot 10^{-142}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V}} \cdot \sqrt{\frac{A}{\ell}}\right)\\ \mathbf{elif}\;V \le 2.480087127913832854241189970085787477828 \cdot 10^{154}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V}} \cdot \sqrt{\frac{A}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))