Average Error: 14.0 → 9.1
Time: 26.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -2.459544082082623877126440069328057953432 \cdot 10^{-105} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 2.475593378518339803291364317838566404249 \cdot 10^{-226}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -\infty:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\

\mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -2.459544082082623877126440069328057953432 \cdot 10^{-105} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 2.475593378518339803291364317838566404249 \cdot 10^{-226}\right):\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r182821 = w0;
        double r182822 = 1.0;
        double r182823 = M;
        double r182824 = D;
        double r182825 = r182823 * r182824;
        double r182826 = 2.0;
        double r182827 = d;
        double r182828 = r182826 * r182827;
        double r182829 = r182825 / r182828;
        double r182830 = pow(r182829, r182826);
        double r182831 = h;
        double r182832 = l;
        double r182833 = r182831 / r182832;
        double r182834 = r182830 * r182833;
        double r182835 = r182822 - r182834;
        double r182836 = sqrt(r182835);
        double r182837 = r182821 * r182836;
        return r182837;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r182838 = M;
        double r182839 = D;
        double r182840 = r182838 * r182839;
        double r182841 = 2.0;
        double r182842 = d;
        double r182843 = r182841 * r182842;
        double r182844 = r182840 / r182843;
        double r182845 = -inf.0;
        bool r182846 = r182844 <= r182845;
        double r182847 = w0;
        double r182848 = 1.0;
        double r182849 = r182843 / r182839;
        double r182850 = r182838 / r182849;
        double r182851 = pow(r182850, r182841);
        double r182852 = h;
        double r182853 = cbrt(r182852);
        double r182854 = r182853 * r182853;
        double r182855 = l;
        double r182856 = cbrt(r182855);
        double r182857 = r182856 * r182856;
        double r182858 = r182854 / r182857;
        double r182859 = r182851 * r182858;
        double r182860 = r182853 / r182856;
        double r182861 = r182859 * r182860;
        double r182862 = r182848 - r182861;
        double r182863 = sqrt(r182862);
        double r182864 = r182847 * r182863;
        double r182865 = -2.459544082082624e-105;
        bool r182866 = r182844 <= r182865;
        double r182867 = 2.4755933785183398e-226;
        bool r182868 = r182844 <= r182867;
        double r182869 = !r182868;
        bool r182870 = r182866 || r182869;
        double r182871 = 2.0;
        double r182872 = r182841 / r182871;
        double r182873 = pow(r182844, r182872);
        double r182874 = r182852 / r182855;
        double r182875 = r182873 * r182874;
        double r182876 = r182873 * r182875;
        double r182877 = r182848 - r182876;
        double r182878 = sqrt(r182877);
        double r182879 = r182847 * r182878;
        double r182880 = sqrt(r182848);
        double r182881 = r182847 * r182880;
        double r182882 = r182870 ? r182879 : r182881;
        double r182883 = r182846 ? r182864 : r182882;
        return r182883;
}

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 3 regimes

  2. if (/ (* M D) (* 2.0 d)) < -inf.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. Using strategy rm

    3. Applied add-cube-cbrt64.0

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

    4. Applied add-cube-cbrt64.0

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

    5. Applied times-frac64.0

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

    6. Applied associate-*r*64.0

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

    8. Applied associate-/l*56.3

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

    if -inf.0 < (/ (* M D) (* 2.0 d)) < -2.459544082082624e-105 or 2.4755933785183398e-226 < (/ (* M D) (* 2.0 d))

    1. Initial program 17.3

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

    3. Applied sqr-pow17.3

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

    4. Applied associate-*l*14.3

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

    if -2.459544082082624e-105 < (/ (* M D) (* 2.0 d)) < 2.4755933785183398e-226

    1. Initial program 7.5

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

    2. Taylor expanded around 0 1.0

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -2.459544082082623877126440069328057953432 \cdot 10^{-105} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 2.475593378518339803291364317838566404249 \cdot 10^{-226}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))))