Henrywood and Agarwal, Equation (3)

Average Error: 18.8 → 1.2

Time: 12.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \le 0.0:\\ \;\;\;\;\left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|\right) \cdot \left(\sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \le 6.81642197495795667 \cdot 10^{304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)\right)}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \end{array}\]

C

double f(double c0, double A, double V, double l) {
        double r205849 = c0;
        double r205850 = A;
        double r205851 = V;
        double r205852 = l;
        double r205853 = r205851 * r205852;
        double r205854 = r205850 / r205853;
        double r205855 = sqrt(r205854);
        double r205856 = r205849 * r205855;
        return r205856;
}

↓

double f(double c0, double A, double V, double l) {
        double r205857 = A;
        double r205858 = V;
        double r205859 = l;
        double r205860 = r205858 * r205859;
        double r205861 = r205857 / r205860;
        double r205862 = 0.0;
        bool r205863 = r205861 <= r205862;
        double r205864 = c0;
        double r205865 = cbrt(r205857);
        double r205866 = cbrt(r205858);
        double r205867 = r205865 / r205866;
        double r205868 = fabs(r205867);
        double r205869 = r205864 * r205868;
        double r205870 = r205865 * r205865;
        double r205871 = cbrt(r205870);
        double r205872 = cbrt(r205866);
        double r205873 = r205872 * r205872;
        double r205874 = r205871 / r205873;
        double r205875 = cbrt(r205859);
        double r205876 = r205875 * r205875;
        double r205877 = r205874 / r205876;
        double r205878 = sqrt(r205877);
        double r205879 = cbrt(r205865);
        double r205880 = r205879 / r205872;
        double r205881 = r205880 / r205875;
        double r205882 = sqrt(r205881);
        double r205883 = r205878 * r205882;
        double r205884 = r205869 * r205883;
        double r205885 = 6.816421974957957e+304;
        bool r205886 = r205861 <= r205885;
        double r205887 = sqrt(r205861);
        double r205888 = r205864 * r205887;
        double r205889 = sqrt(r205874);
        double r205890 = r205889 * r205882;
        double r205891 = r205868 * r205890;
        double r205892 = r205864 * r205891;
        double r205893 = sqrt(r205876);
        double r205894 = r205892 / r205893;
        double r205895 = r205886 ? r205888 : r205894;
        double r205896 = r205863 ? r205884 : r205895;
        return r205896;
}

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Derivation

1. Split input into 3 regimes

2. if (/ A (* V l)) < 0.0

   1. Initial program 41.0

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

   3. Applied associate-/r* 29.8

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

   5. Applied *-un-lft-identity 29.8

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

   6. Applied add-cube-cbrt 29.9

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

   7. Applied add-cube-cbrt 29.9

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

   8. Applied times-frac 29.9

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

   9. Applied times-frac 29.9

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

   10. Applied sqrt-prod 10.1

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

   11. Simplified 7.8

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

   13. Applied add-cube-cbrt 7.8

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

   14. Applied add-cube-cbrt 7.9

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

   15. Applied add-cube-cbrt 7.9

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

   16. Applied cbrt-prod 7.9

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

   17. Applied times-frac 7.9

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

   18. Applied times-frac 7.9

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

   19. Applied sqrt-prod 2.9

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

   21. Applied associate-*r* 1.3

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

   if 0.0 < (/ A (* V l)) < 6.816421974957957e+304

   1. Initial program 0.8

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

   3. Applied associate-/r* 8.7

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

   4. Taylor expanded around 0 0.8

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

   if 6.816421974957957e+304 < (/ A (* V l))

   1. Initial program 63.0

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

   3. Applied associate-/r* 48.0

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

   5. Applied *-un-lft-identity 48.0

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

   6. Applied add-cube-cbrt 48.2

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

   7. Applied add-cube-cbrt 48.2

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

   8. Applied times-frac 48.2

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

   9. Applied times-frac 47.9

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

   10. Applied sqrt-prod 20.1

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

   11. Simplified 15.0

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

   13. Applied add-cube-cbrt 15.2

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

   14. Applied add-cube-cbrt 15.3

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

   15. Applied add-cube-cbrt 15.3

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

   16. Applied cbrt-prod 15.3

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

   17. Applied times-frac 15.3

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

   18. Applied times-frac 15.3

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

   19. Applied sqrt-prod 5.3

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

   21. Applied sqrt-div 5.3

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

   22. Applied associate-*l/ 5.3

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

   23. Applied associate-*r/ 5.3

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

   24. Applied associate-*r/ 2.4

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \le 0.0:\\ \;\;\;\;\left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|\right) \cdot \left(\sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \le 6.81642197495795667 \cdot 10^{304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)\right)}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))