Average Error: 14.2 → 9.7
Time: 23.1s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.769921175136561925478600486681328667335 \cdot 10^{308}:\\ \;\;\;\;\left(w0 \cdot \left(\sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}} \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\right)\right) \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.769921175136561925478600486681328667335 \cdot 10^{308}:\\
\;\;\;\;\left(w0 \cdot \left(\sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}} \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\right)\right) \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{1} \cdot w0\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r135862 = w0;
        double r135863 = 1.0;
        double r135864 = M;
        double r135865 = D;
        double r135866 = r135864 * r135865;
        double r135867 = 2.0;
        double r135868 = d;
        double r135869 = r135867 * r135868;
        double r135870 = r135866 / r135869;
        double r135871 = pow(r135870, r135867);
        double r135872 = h;
        double r135873 = l;
        double r135874 = r135872 / r135873;
        double r135875 = r135871 * r135874;
        double r135876 = r135863 - r135875;
        double r135877 = sqrt(r135876);
        double r135878 = r135862 * r135877;
        return r135878;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r135879 = M;
        double r135880 = D;
        double r135881 = r135879 * r135880;
        double r135882 = 2.0;
        double r135883 = d;
        double r135884 = r135882 * r135883;
        double r135885 = r135881 / r135884;
        double r135886 = pow(r135885, r135882);
        double r135887 = 1.769921175136562e+308;
        bool r135888 = r135886 <= r135887;
        double r135889 = w0;
        double r135890 = 1.0;
        double r135891 = h;
        double r135892 = r135886 * r135891;
        double r135893 = l;
        double r135894 = cbrt(r135893);
        double r135895 = r135894 * r135894;
        double r135896 = r135892 / r135895;
        double r135897 = 1.0;
        double r135898 = cbrt(r135897);
        double r135899 = r135898 / r135894;
        double r135900 = r135896 * r135899;
        double r135901 = r135890 - r135900;
        double r135902 = sqrt(r135901);
        double r135903 = cbrt(r135902);
        double r135904 = r135903 * r135903;
        double r135905 = r135889 * r135904;
        double r135906 = r135905 * r135903;
        double r135907 = sqrt(r135890);
        double r135908 = r135907 * r135889;
        double r135909 = r135888 ? r135906 : r135908;
        return r135909;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (pow (/ (* M D) (* 2.0 d)) 2.0) < 1.769921175136562e+308

    1. Initial program 7.0

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied div-inv7.0

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]

    4. Applied associate-*r*3.1

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt3.2

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\]

    7. Applied add-cube-cbrt3.2

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

    8. Applied times-frac3.2

      \[\leadsto w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}\right)}}\]

    9. Applied associate-*r*3.2

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\]

    10. Simplified3.2

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt3.2

      \[\leadsto w0 \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}} \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\right) \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\right)}\]

    13. Applied associate-*r*3.2

      \[\leadsto \color{blue}{\left(w0 \cdot \left(\sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}} \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\right)\right) \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}}\]

    if 1.769921175136562e+308 < (pow (/ (* M D) (* 2.0 d)) 2.0)

    1. Initial program 64.0

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

    2. Taylor expanded around 0 54.0

      \[\leadsto \color{blue}{\sqrt{1} \cdot w0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.769921175136561925478600486681328667335 \cdot 10^{308}:\\ \;\;\;\;\left(w0 \cdot \left(\sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}} \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\right)\right) \cdot \sqrt[3]{\sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))