Average Error: 14.2 → 8.9
Time: 11.4s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.06029486315833535 \cdot 10^{139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -4.41799484384605494 \cdot 10^{-176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \le -3.06029486315833535 \cdot 10^{139}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -4.41799484384605494 \cdot 10^{-176}:\\
\;\;\;\;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 - \frac{1}{\frac{\ell}{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}}}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r220817 = w0;
        double r220818 = 1.0;
        double r220819 = M;
        double r220820 = D;
        double r220821 = r220819 * r220820;
        double r220822 = 2.0;
        double r220823 = d;
        double r220824 = r220822 * r220823;
        double r220825 = r220821 / r220824;
        double r220826 = pow(r220825, r220822);
        double r220827 = h;
        double r220828 = l;
        double r220829 = r220827 / r220828;
        double r220830 = r220826 * r220829;
        double r220831 = r220818 - r220830;
        double r220832 = sqrt(r220831);
        double r220833 = r220817 * r220832;
        return r220833;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r220834 = h;
        double r220835 = l;
        double r220836 = r220834 / r220835;
        double r220837 = -3.0602948631583354e+139;
        bool r220838 = r220836 <= r220837;
        double r220839 = w0;
        double r220840 = 1.0;
        double r220841 = 1.0;
        double r220842 = M;
        double r220843 = D;
        double r220844 = r220842 * r220843;
        double r220845 = 2.0;
        double r220846 = d;
        double r220847 = r220845 * r220846;
        double r220848 = r220844 / r220847;
        double r220849 = pow(r220848, r220845);
        double r220850 = r220849 * r220834;
        double r220851 = r220835 / r220850;
        double r220852 = r220841 / r220851;
        double r220853 = r220840 - r220852;
        double r220854 = sqrt(r220853);
        double r220855 = r220839 * r220854;
        double r220856 = -4.417994843846055e-176;
        bool r220857 = r220836 <= r220856;
        double r220858 = 2.0;
        double r220859 = r220845 / r220858;
        double r220860 = pow(r220848, r220859);
        double r220861 = r220860 * r220836;
        double r220862 = r220860 * r220861;
        double r220863 = r220840 - r220862;
        double r220864 = sqrt(r220863);
        double r220865 = r220839 * r220864;
        double r220866 = r220847 / r220843;
        double r220867 = r220842 / r220866;
        double r220868 = pow(r220867, r220859);
        double r220869 = r220868 * r220834;
        double r220870 = r220868 * r220869;
        double r220871 = r220835 / r220870;
        double r220872 = r220841 / r220871;
        double r220873 = r220840 - r220872;
        double r220874 = sqrt(r220873);
        double r220875 = r220839 * r220874;
        double r220876 = r220857 ? r220865 : r220875;
        double r220877 = r220838 ? r220855 : r220876;
        return r220877;
}

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 (/ h l) < -3.0602948631583354e+139

    1. Initial program 34.4

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

    3. Applied associate-*r/19.9

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

    5. Applied clear-num19.9

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

    if -3.0602948631583354e+139 < (/ h l) < -4.417994843846055e-176

    1. Initial program 14.2

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

    3. Applied sqr-pow14.2

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

    4. Applied associate-*l*12.7

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

    if -4.417994843846055e-176 < (/ h l)

    1. Initial program 8.6

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

    3. Applied associate-*r/5.8

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

    5. Applied clear-num5.8

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

    7. Applied associate-/l*5.7

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

    9. Applied sqr-pow5.7

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

    10. Applied associate-*l*3.9

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.06029486315833535 \cdot 10^{139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -4.41799484384605494 \cdot 10^{-176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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))))))