Average Error: 18.9 → 11.0
Time: 6.4s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \le -7.24944730302796452 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)} \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\right) \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\\ \mathbf{elif}\;V \le -3.2698369089058447 \cdot 10^{-220}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \le 8.37998221063094693 \cdot 10^{-237}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{\sqrt[3]{A}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{V}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)\right)\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \le -7.24944730302796452 \cdot 10^{-9}:\\
\;\;\;\;\left(\sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)} \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\right) \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\\

\mathbf{elif}\;V \le -3.2698369089058447 \cdot 10^{-220}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\

\mathbf{elif}\;V \le 8.37998221063094693 \cdot 10^{-237}:\\
\;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{\sqrt[3]{A}}}}\right)\\

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

\end{array}
double f(double c0, double A, double V, double l) {
        double r242585 = c0;
        double r242586 = A;
        double r242587 = V;
        double r242588 = l;
        double r242589 = r242587 * r242588;
        double r242590 = r242586 / r242589;
        double r242591 = sqrt(r242590);
        double r242592 = r242585 * r242591;
        return r242592;
}
↓
double f(double c0, double A, double V, double l) {
        double r242593 = V;
        double r242594 = -7.2494473030279645e-09;
        bool r242595 = r242593 <= r242594;
        double r242596 = c0;
        double r242597 = A;
        double r242598 = cbrt(r242597);
        double r242599 = fabs(r242598);
        double r242600 = l;
        double r242601 = r242593 * r242600;
        double r242602 = r242598 / r242601;
        double r242603 = sqrt(r242602);
        double r242604 = r242599 * r242603;
        double r242605 = r242596 * r242604;
        double r242606 = cbrt(r242605);
        double r242607 = r242606 * r242606;
        double r242608 = r242607 * r242606;
        double r242609 = -3.2698369089058447e-220;
        bool r242610 = r242593 <= r242609;
        double r242611 = r242597 / r242593;
        double r242612 = r242611 / r242600;
        double r242613 = sqrt(r242612);
        double r242614 = r242596 * r242613;
        double r242615 = 8.379982210630947e-237;
        bool r242616 = r242593 <= r242615;
        double r242617 = 1.0;
        double r242618 = r242601 / r242598;
        double r242619 = r242617 / r242618;
        double r242620 = sqrt(r242619);
        double r242621 = r242599 * r242620;
        double r242622 = r242596 * r242621;
        double r242623 = r242598 * r242598;
        double r242624 = cbrt(r242623);
        double r242625 = r242624 / r242593;
        double r242626 = sqrt(r242625);
        double r242627 = cbrt(r242598);
        double r242628 = r242627 / r242600;
        double r242629 = sqrt(r242628);
        double r242630 = r242626 * r242629;
        double r242631 = r242599 * r242630;
        double r242632 = r242596 * r242631;
        double r242633 = r242616 ? r242622 : r242632;
        double r242634 = r242610 ? r242614 : r242633;
        double r242635 = r242595 ? r242608 : r242634;
        return r242635;
}

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 < -7.2494473030279645e-09

    1. Initial program 18.1

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

    3. Applied add-cube-cbrt18.4

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

    4. Applied associate-/l*18.4

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

    6. Applied div-inv18.4

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

    7. Applied sqrt-prod13.4

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

    8. Simplified13.4

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

    9. Simplified13.1

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

    11. Applied add-cube-cbrt13.5

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

    if -7.2494473030279645e-09 < V < -3.2698369089058447e-220

    1. Initial program 17.5

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

    3. Applied associate-/r*18.1

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

    if -3.2698369089058447e-220 < V < 8.379982210630947e-237

    1. Initial program 30.0

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

    3. Applied add-cube-cbrt30.3

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

    4. Applied associate-/l*30.3

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

    6. Applied div-inv30.4

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

    7. Applied sqrt-prod25.8

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

    8. Simplified25.8

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

    9. Simplified25.8

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

    11. Applied clear-num25.8

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

    if 8.379982210630947e-237 < V

    1. Initial program 16.9

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

    3. Applied add-cube-cbrt17.3

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

    4. Applied associate-/l*17.3

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

    6. Applied div-inv17.3

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

    7. Applied sqrt-prod11.9

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

    8. Simplified11.9

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

    9. Simplified11.6

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

    11. Applied add-cube-cbrt11.7

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

    12. Applied cbrt-prod11.7

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

    13. Applied times-frac10.3

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

    14. Applied sqrt-prod3.0

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \le -7.24944730302796452 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)} \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\right) \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\\ \mathbf{elif}\;V \le -3.2698369089058447 \cdot 10^{-220}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \le 8.37998221063094693 \cdot 10^{-237}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{\sqrt[3]{A}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{V}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))