Average Error: 19.0 → 7.8
Time: 6.4s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.8700420056280899 \cdot 10^{68}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \le -3.66804977381144 \cdot 10^{-167}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \frac{\sqrt[3]{A}}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \mathbf{elif}\;V \cdot \ell \le 2.01505104436057934 \cdot 10^{305}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \cdot \ell \le -1.8700420056280899 \cdot 10^{68}:\\
\;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\

\mathbf{elif}\;V \cdot \ell \le -3.66804977381144 \cdot 10^{-167}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\

\mathbf{elif}\;V \cdot \ell \le 0.0:\\
\;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \frac{\sqrt[3]{A}}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\

\mathbf{elif}\;V \cdot \ell \le 2.01505104436057934 \cdot 10^{305}:\\
\;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\

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

\end{array}
double f(double c0, double A, double V, double l) {
        double r147873 = c0;
        double r147874 = A;
        double r147875 = V;
        double r147876 = l;
        double r147877 = r147875 * r147876;
        double r147878 = r147874 / r147877;
        double r147879 = sqrt(r147878);
        double r147880 = r147873 * r147879;
        return r147880;
}
↓
double f(double c0, double A, double V, double l) {
        double r147881 = V;
        double r147882 = l;
        double r147883 = r147881 * r147882;
        double r147884 = -1.87004200562809e+68;
        bool r147885 = r147883 <= r147884;
        double r147886 = c0;
        double r147887 = A;
        double r147888 = cbrt(r147887);
        double r147889 = r147888 * r147888;
        double r147890 = r147889 / r147881;
        double r147891 = cbrt(r147890);
        double r147892 = fabs(r147891);
        double r147893 = r147888 / r147882;
        double r147894 = r147891 * r147893;
        double r147895 = sqrt(r147894);
        double r147896 = r147892 * r147895;
        double r147897 = r147886 * r147896;
        double r147898 = -3.66804977381144e-167;
        bool r147899 = r147883 <= r147898;
        double r147900 = 1.0;
        double r147901 = r147883 / r147887;
        double r147902 = r147900 / r147901;
        double r147903 = sqrt(r147902);
        double r147904 = r147886 * r147903;
        double r147905 = 0.0;
        bool r147906 = r147883 <= r147905;
        double r147907 = cbrt(r147889);
        double r147908 = r147907 * r147891;
        double r147909 = r147907 * r147893;
        double r147910 = r147908 * r147909;
        double r147911 = sqrt(r147910);
        double r147912 = cbrt(r147881);
        double r147913 = fabs(r147912);
        double r147914 = r147911 / r147913;
        double r147915 = r147886 * r147914;
        double r147916 = 2.0150510443605793e+305;
        bool r147917 = r147883 <= r147916;
        double r147918 = sqrt(r147887);
        double r147919 = r147900 / r147883;
        double r147920 = sqrt(r147919);
        double r147921 = r147918 * r147920;
        double r147922 = r147886 * r147921;
        double r147923 = r147917 ? r147922 : r147897;
        double r147924 = r147906 ? r147915 : r147923;
        double r147925 = r147899 ? r147904 : r147924;
        double r147926 = r147885 ? r147897 : r147925;
        return r147926;
}

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 l) < -1.87004200562809e+68 or 2.0150510443605793e+305 < (* V l)

    1. Initial program 25.7

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

    3. Applied add-cube-cbrt25.9

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

    4. Applied times-frac17.9

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

    6. Applied add-cube-cbrt18.0

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

    7. Applied associate-*l*18.0

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

    9. Applied sqrt-prod11.7

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

    10. Simplified11.7

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

    if -1.87004200562809e+68 < (* V l) < -3.66804977381144e-167

    1. Initial program 5.0

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

    3. Applied clear-num5.1

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

    if -3.66804977381144e-167 < (* V l) < 0.0

    1. Initial program 43.9

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

    3. Applied add-cube-cbrt44.1

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

    4. Applied times-frac30.2

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

    6. Applied add-cube-cbrt30.4

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

    7. Applied associate-*l*30.4

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

    9. Applied cbrt-div30.3

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

    10. Applied associate-*l/30.4

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

    11. Applied cbrt-div30.4

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

    12. Applied associate-*r/30.4

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

    13. Applied frac-times30.5

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

    14. Applied sqrt-div19.9

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

    15. Simplified19.9

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

    16. Simplified19.9

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

    if 0.0 < (* V l) < 2.0150510443605793e+305

    1. Initial program 10.2

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

    3. Applied div-inv10.6

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

    4. Applied sqrt-prod1.4

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.8700420056280899 \cdot 10^{68}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \le -3.66804977381144 \cdot 10^{-167}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \frac{\sqrt[3]{A}}{\ell}\right)}}{\left|\sqrt[3]{V}\right|}\\ \mathbf{elif}\;V \cdot \ell \le 2.01505104436057934 \cdot 10^{305}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right| \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \frac{\sqrt[3]{A}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))