Average Error: 19.3 → 4.0
Time: 23.0s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le 8.30925288092755745715389309116932962946 \cdot 10^{-314} \lor \neg \left(V \cdot \ell \le 9.368575440665084813070724399515047023084 \cdot 10^{243}\right):\\ \;\;\;\;\left(\frac{\left|\frac{\sqrt[3]{\frac{A}{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}\right|}{\sqrt[3]{\left|\sqrt[3]{V}\right|} \cdot \sqrt[3]{\left|\sqrt[3]{V}\right|}} \cdot c0\right) \cdot \frac{\sqrt{\frac{\sqrt[3]{\frac{A}{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\left|\sqrt[3]{V}\right|}}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot \sqrt{A}\right) \cdot \sqrt{\frac{1}{V \cdot \ell}}\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \cdot \ell \le 8.30925288092755745715389309116932962946 \cdot 10^{-314} \lor \neg \left(V \cdot \ell \le 9.368575440665084813070724399515047023084 \cdot 10^{243}\right):\\
\;\;\;\;\left(\frac{\left|\frac{\sqrt[3]{\frac{A}{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}\right|}{\sqrt[3]{\left|\sqrt[3]{V}\right|} \cdot \sqrt[3]{\left|\sqrt[3]{V}\right|}} \cdot c0\right) \cdot \frac{\sqrt{\frac{\sqrt[3]{\frac{A}{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\left|\sqrt[3]{V}\right|}}\\

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

\end{array}
double f(double c0, double A, double V, double l) {
        double r109808 = c0;
        double r109809 = A;
        double r109810 = V;
        double r109811 = l;
        double r109812 = r109810 * r109811;
        double r109813 = r109809 / r109812;
        double r109814 = sqrt(r109813);
        double r109815 = r109808 * r109814;
        return r109815;
}
↓
double f(double c0, double A, double V, double l) {
        double r109816 = V;
        double r109817 = l;
        double r109818 = r109816 * r109817;
        double r109819 = 8.3092528809276e-314;
        bool r109820 = r109818 <= r109819;
        double r109821 = 9.368575440665085e+243;
        bool r109822 = r109818 <= r109821;
        double r109823 = !r109822;
        bool r109824 = r109820 || r109823;
        double r109825 = A;
        double r109826 = cbrt(r109816);
        double r109827 = r109825 / r109826;
        double r109828 = cbrt(r109827);
        double r109829 = cbrt(r109817);
        double r109830 = r109828 / r109829;
        double r109831 = fabs(r109830);
        double r109832 = fabs(r109826);
        double r109833 = cbrt(r109832);
        double r109834 = r109833 * r109833;
        double r109835 = r109831 / r109834;
        double r109836 = c0;
        double r109837 = r109835 * r109836;
        double r109838 = sqrt(r109830);
        double r109839 = r109838 / r109833;
        double r109840 = r109837 * r109839;
        double r109841 = sqrt(r109825);
        double r109842 = r109836 * r109841;
        double r109843 = 1.0;
        double r109844 = r109843 / r109818;
        double r109845 = sqrt(r109844);
        double r109846 = r109842 * r109845;
        double r109847 = r109824 ? r109840 : r109846;
        return r109847;
}

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 (* V l) < 8.3092528809276e-314 or 9.368575440665085e+243 < (* V l)

    1. Initial program 24.3

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

    3. Applied add-cube-cbrt24.6

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

    4. Applied times-frac20.1

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

    6. Applied add-cube-cbrt20.2

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

    7. Applied times-frac20.2

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

    8. Applied associate-*l*18.3

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

    10. Applied associate-*l/19.4

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

    11. Applied sqrt-div12.9

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

    12. Simplified14.0

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

    13. Simplified14.0

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

    15. Applied add-cube-cbrt14.4

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

    16. Applied add-cube-cbrt14.5

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

    17. Applied add-cube-cbrt14.5

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

    18. Applied times-frac14.5

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

    19. Applied sqrt-prod6.8

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

    20. Applied times-frac6.8

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

    21. Applied associate-*r*6.7

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

    22. Simplified4.7

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

    if 8.3092528809276e-314 < (* V l) < 9.368575440665085e+243

    1. Initial program 10.1

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

    3. Applied div-inv10.2

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

    4. Applied sqrt-prod0.6

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

    5. Applied associate-*r*2.8

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

    6. Simplified2.8

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le 8.30925288092755745715389309116932962946 \cdot 10^{-314} \lor \neg \left(V \cdot \ell \le 9.368575440665084813070724399515047023084 \cdot 10^{243}\right):\\ \;\;\;\;\left(\frac{\left|\frac{\sqrt[3]{\frac{A}{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}\right|}{\sqrt[3]{\left|\sqrt[3]{V}\right|} \cdot \sqrt[3]{\left|\sqrt[3]{V}\right|}} \cdot c0\right) \cdot \frac{\sqrt{\frac{\sqrt[3]{\frac{A}{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\left|\sqrt[3]{V}\right|}}\\ \mathbf{else}:\\ \;\;\;\;\left(c0 \cdot \sqrt{A}\right) \cdot \sqrt{\frac{1}{V \cdot \ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))