Average Error: 18.9 → 11.0
Time: 6.2s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \le -7.24944730302796452 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)} \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\right) \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\\ \mathbf{elif}\;V \le -3.2698369089058447 \cdot 10^{-220}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \le 8.37998221063094693 \cdot 10^{-237}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{1}}{\frac{V \cdot \ell}{\sqrt[3]{A}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{V}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)\right)\\ \end{array}\]
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{\sqrt[3]{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 r181716 = c0;
        double r181717 = A;
        double r181718 = V;
        double r181719 = l;
        double r181720 = r181718 * r181719;
        double r181721 = r181717 / r181720;
        double r181722 = sqrt(r181721);
        double r181723 = r181716 * r181722;
        return r181723;
}
↓
double f(double c0, double A, double V, double l) {
        double r181724 = V;
        double r181725 = -7.2494473030279645e-09;
        bool r181726 = r181724 <= r181725;
        double r181727 = c0;
        double r181728 = A;
        double r181729 = cbrt(r181728);
        double r181730 = fabs(r181729);
        double r181731 = l;
        double r181732 = r181724 * r181731;
        double r181733 = r181729 / r181732;
        double r181734 = sqrt(r181733);
        double r181735 = r181730 * r181734;
        double r181736 = r181727 * r181735;
        double r181737 = cbrt(r181736);
        double r181738 = r181737 * r181737;
        double r181739 = r181738 * r181737;
        double r181740 = -3.2698369089058447e-220;
        bool r181741 = r181724 <= r181740;
        double r181742 = r181728 / r181724;
        double r181743 = r181742 / r181731;
        double r181744 = sqrt(r181743);
        double r181745 = r181727 * r181744;
        double r181746 = 8.379982210630947e-237;
        bool r181747 = r181724 <= r181746;
        double r181748 = 1.0;
        double r181749 = cbrt(r181748);
        double r181750 = r181732 / r181729;
        double r181751 = r181749 / r181750;
        double r181752 = sqrt(r181751);
        double r181753 = r181730 * r181752;
        double r181754 = r181727 * r181753;
        double r181755 = r181729 * r181729;
        double r181756 = cbrt(r181755);
        double r181757 = r181756 / r181724;
        double r181758 = sqrt(r181757);
        double r181759 = cbrt(r181729);
        double r181760 = r181759 / r181731;
        double r181761 = sqrt(r181760);
        double r181762 = r181758 * r181761;
        double r181763 = r181730 * r181762;
        double r181764 = r181727 * r181763;
        double r181765 = r181747 ? r181754 : r181764;
        double r181766 = r181741 ? r181745 : r181765;
        double r181767 = r181726 ? r181739 : r181766;
        return r181767;
}

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 *-un-lft-identity25.8

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

    12. Applied cbrt-prod25.8

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

    13. Applied associate-/l*25.8

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

    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

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

Reproduce

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