Average Error: 19.0 → 7.8
Time: 6.5s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.8700420056280899 \cdot 10^{68}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \le -3.66804977381144 \cdot 10^{-167}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \frac{\sqrt[3]{A}}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \mathbf{elif}\;V \cdot \ell \le 2.01505104436057934 \cdot 10^{305}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \cdot \ell \le -1.8700420056280899 \cdot 10^{68}:\\
\;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\

\mathbf{elif}\;V \cdot \ell \le -3.66804977381144 \cdot 10^{-167}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\

\mathbf{elif}\;V \cdot \ell \le 0.0:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \frac{\sqrt[3]{A}}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\

\mathbf{elif}\;V \cdot \ell \le 2.01505104436057934 \cdot 10^{305}:\\
\;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\

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

\end{array}
double f(double c0, double A, double V, double l) {
        double r134609 = c0;
        double r134610 = A;
        double r134611 = V;
        double r134612 = l;
        double r134613 = r134611 * r134612;
        double r134614 = r134610 / r134613;
        double r134615 = sqrt(r134614);
        double r134616 = r134609 * r134615;
        return r134616;
}
↓
double f(double c0, double A, double V, double l) {
        double r134617 = V;
        double r134618 = l;
        double r134619 = r134617 * r134618;
        double r134620 = -1.87004200562809e+68;
        bool r134621 = r134619 <= r134620;
        double r134622 = c0;
        double r134623 = A;
        double r134624 = cbrt(r134623);
        double r134625 = r134624 * r134624;
        double r134626 = r134625 / r134617;
        double r134627 = cbrt(r134626);
        double r134628 = fabs(r134627);
        double r134629 = r134624 / r134618;
        double r134630 = r134627 * r134629;
        double r134631 = sqrt(r134630);
        double r134632 = r134628 * r134631;
        double r134633 = r134622 * r134632;
        double r134634 = -3.66804977381144e-167;
        bool r134635 = r134619 <= r134634;
        double r134636 = 1.0;
        double r134637 = r134619 / r134623;
        double r134638 = r134636 / r134637;
        double r134639 = sqrt(r134638);
        double r134640 = r134622 * r134639;
        double r134641 = 0.0;
        bool r134642 = r134619 <= r134641;
        double r134643 = cbrt(r134625);
        double r134644 = r134643 * r134627;
        double r134645 = r134643 * r134629;
        double r134646 = r134644 * r134645;
        double r134647 = sqrt(r134646);
        double r134648 = cbrt(r134617);
        double r134649 = fabs(r134648);
        double r134650 = r134647 / r134649;
        double r134651 = r134622 * r134650;
        double r134652 = 2.0150510443605793e+305;
        bool r134653 = r134619 <= r134652;
        double r134654 = sqrt(r134623);
        double r134655 = r134636 / r134619;
        double r134656 = sqrt(r134655);
        double r134657 = r134654 * r134656;
        double r134658 = r134622 * r134657;
        double r134659 = r134653 ? r134658 : r134633;
        double r134660 = r134642 ? r134651 : r134659;
        double r134661 = r134635 ? r134640 : r134660;
        double r134662 = r134621 ? r134633 : r134661;
        return r134662;
}

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 4 regimes

  2. if (* V l) < -1.87004200562809e+68 or 2.0150510443605793e+305 < (* V l)

    1. Initial program 25.7

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

    3. Applied add-cube-cbrt25.9

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

    4. Applied times-frac17.9

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

    6. Applied add-cube-cbrt18.0

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

    7. Applied associate-*l*18.0

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

    9. Applied sqrt-prod11.7

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

    10. Simplified11.7

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

    if -1.87004200562809e+68 < (* V l) < -3.66804977381144e-167

    1. Initial program 5.0

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

    3. Applied clear-num5.1

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

    if -3.66804977381144e-167 < (* V l) < 0.0

    1. Initial program 43.9

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

    3. Applied add-cube-cbrt44.1

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

    4. Applied times-frac30.2

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

    6. Applied add-cube-cbrt30.4

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

    7. Applied associate-*l*30.4

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

    9. Applied cbrt-div30.3

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

    10. Applied associate-*l/30.4

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

    11. Applied cbrt-div30.4

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

    12. Applied associate-*r/30.4

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

    13. Applied frac-times30.5

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

    14. Applied sqrt-div19.9

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

    15. Simplified19.9

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

    16. Simplified19.9

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

    if 0.0 < (* V l) < 2.0150510443605793e+305

    1. Initial program 10.2

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

    3. Applied div-inv10.6

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

    4. Applied sqrt-prod1.4

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.8700420056280899 \cdot 10^{68}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \le -3.66804977381144 \cdot 10^{-167}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \frac{\sqrt[3]{A}}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \mathbf{elif}\;V \cdot \ell \le 2.01505104436057934 \cdot 10^{305}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))