Average Error: 19.1 → 0.8
Time: 36.6s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \le 4.189537493318518753431675781489977417082 \cdot 10^{-310}:\\ \;\;\;\;\left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right|\right) \cdot \left(\sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{\ell}}}}{\sqrt[3]{V}}}\right)\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \le 8.02558393739106920381790045636343785553 \cdot 10^{288}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\sqrt[3]{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\right) \cdot \sqrt{\frac{\sqrt[3]{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}{\sqrt[3]{V}}}\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{A}{V \cdot \ell} \le 4.189537493318518753431675781489977417082 \cdot 10^{-310}:\\
\;\;\;\;\left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right|\right) \cdot \left(\sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{\ell}}}}{\sqrt[3]{V}}}\right)\\

\mathbf{elif}\;\frac{A}{V \cdot \ell} \le 8.02558393739106920381790045636343785553 \cdot 10^{288}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\

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

\end{array}
double f(double c0, double A, double V, double l) {
        double r101936 = c0;
        double r101937 = A;
        double r101938 = V;
        double r101939 = l;
        double r101940 = r101938 * r101939;
        double r101941 = r101937 / r101940;
        double r101942 = sqrt(r101941);
        double r101943 = r101936 * r101942;
        return r101943;
}
↓
double f(double c0, double A, double V, double l) {
        double r101944 = A;
        double r101945 = V;
        double r101946 = l;
        double r101947 = r101945 * r101946;
        double r101948 = r101944 / r101947;
        double r101949 = 4.1895374933185e-310;
        bool r101950 = r101948 <= r101949;
        double r101951 = c0;
        double r101952 = cbrt(r101944);
        double r101953 = cbrt(r101946);
        double r101954 = r101952 / r101953;
        double r101955 = fabs(r101954);
        double r101956 = r101951 * r101955;
        double r101957 = r101952 * r101952;
        double r101958 = cbrt(r101957);
        double r101959 = r101953 * r101953;
        double r101960 = cbrt(r101959);
        double r101961 = r101958 / r101960;
        double r101962 = cbrt(r101945);
        double r101963 = r101962 * r101962;
        double r101964 = r101961 / r101963;
        double r101965 = sqrt(r101964);
        double r101966 = cbrt(r101952);
        double r101967 = cbrt(r101953);
        double r101968 = r101966 / r101967;
        double r101969 = r101968 / r101962;
        double r101970 = sqrt(r101969);
        double r101971 = r101965 * r101970;
        double r101972 = r101956 * r101971;
        double r101973 = 8.025583937391069e+288;
        bool r101974 = r101948 <= r101973;
        double r101975 = sqrt(r101948);
        double r101976 = r101951 * r101975;
        double r101977 = cbrt(r101954);
        double r101978 = r101977 * r101977;
        double r101979 = r101978 / r101963;
        double r101980 = sqrt(r101979);
        double r101981 = r101956 * r101980;
        double r101982 = r101977 / r101962;
        double r101983 = sqrt(r101982);
        double r101984 = r101981 * r101983;
        double r101985 = r101974 ? r101976 : r101984;
        double r101986 = r101950 ? r101972 : r101985;
        return r101986;
}

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 3 regimes

  2. if (/ A (* V l)) < 4.1895374933185e-310

    1. Initial program 41.4

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

    3. Applied *-un-lft-identity41.4

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

    4. Applied times-frac31.5

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\]
    5. Using strategy rm

    6. Applied associate-*l/31.5

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

    7. Simplified31.5

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

    9. Applied *-un-lft-identity31.5

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

    10. Applied add-cube-cbrt31.6

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

    11. Applied add-cube-cbrt31.6

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

    12. Applied times-frac31.6

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

    13. Applied times-frac31.3

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

    14. Applied sqrt-prod11.2

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

    15. Applied associate-*r*11.5

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

    16. Simplified8.5

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

    18. Applied add-cube-cbrt8.6

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

    19. Applied add-cube-cbrt8.6

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

    20. Applied cbrt-prod8.6

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

    21. Applied add-cube-cbrt8.6

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

    22. Applied cbrt-prod8.6

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

    23. Applied times-frac8.6

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

    24. Applied times-frac8.6

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

    25. Applied sqrt-prod1.3

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

    if 4.1895374933185e-310 < (/ A (* V l)) < 8.025583937391069e+288

    1. Initial program 0.4

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

    if 8.025583937391069e+288 < (/ A (* V l))

    1. Initial program 59.7

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

    3. Applied *-un-lft-identity59.7

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

    4. Applied times-frac47.8

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\]
    5. Using strategy rm

    6. Applied associate-*l/47.8

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

    7. Simplified47.8

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

    9. Applied *-un-lft-identity47.8

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

    10. Applied add-cube-cbrt47.9

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

    11. Applied add-cube-cbrt48.0

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

    12. Applied times-frac48.0

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

    13. Applied times-frac47.1

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

    14. Applied sqrt-prod17.9

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

    15. Applied associate-*r*18.1

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

    16. Simplified14.0

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

    18. Applied add-cube-cbrt14.1

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

    19. Applied add-cube-cbrt14.2

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

    20. Applied times-frac14.2

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

    21. Applied sqrt-prod1.7

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

    22. Applied associate-*r*1.7

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \le 4.189537493318518753431675781489977417082 \cdot 10^{-310}:\\ \;\;\;\;\left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right|\right) \cdot \left(\sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{\ell}}}}{\sqrt[3]{V}}}\right)\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \le 8.02558393739106920381790045636343785553 \cdot 10^{288}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\sqrt[3]{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\right) \cdot \sqrt{\frac{\sqrt[3]{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}{\sqrt[3]{V}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))