Average Error: 14.4 → 10.9
Time: 16.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.677179237977214472415692974466936692272 \cdot 10^{305} \lor \frac{h}{\ell} \le \frac{-77024497644807}{5.109351192408882717840314145374306729133 \cdot 10^{294}}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \le -4.677179237977214472415692974466936692272 \cdot 10^{305} \lor \frac{h}{\ell} \le \frac{-77024497644807}{5.109351192408882717840314145374306729133 \cdot 10^{294}}:\\
\;\;\;\;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 r169827 = w0;
        double r169828 = 1.0;
        double r169829 = M;
        double r169830 = D;
        double r169831 = r169829 * r169830;
        double r169832 = 2.0;
        double r169833 = d;
        double r169834 = r169832 * r169833;
        double r169835 = r169831 / r169834;
        double r169836 = pow(r169835, r169832);
        double r169837 = h;
        double r169838 = l;
        double r169839 = r169837 / r169838;
        double r169840 = r169836 * r169839;
        double r169841 = r169828 - r169840;
        double r169842 = sqrt(r169841);
        double r169843 = r169827 * r169842;
        return r169843;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r169844 = h;
        double r169845 = l;
        double r169846 = r169844 / r169845;
        double r169847 = -4.6771792379772145e+305;
        bool r169848 = r169846 <= r169847;
        double r169849 = -77024497644807.0;
        double r169850 = 5.109351192408883e+294;
        double r169851 = r169849 / r169850;
        bool r169852 = r169846 <= r169851;
        bool r169853 = r169848 || r169852;
        double r169854 = w0;
        double r169855 = 1.0;
        double r169856 = M;
        double r169857 = D;
        double r169858 = r169856 * r169857;
        double r169859 = 2.0;
        double r169860 = d;
        double r169861 = r169859 * r169860;
        double r169862 = r169858 / r169861;
        double r169863 = 2.0;
        double r169864 = r169859 / r169863;
        double r169865 = pow(r169862, r169864);
        double r169866 = r169865 * r169846;
        double r169867 = r169865 * r169866;
        double r169868 = r169855 - r169867;
        double r169869 = sqrt(r169868);
        double r169870 = r169854 * r169869;
        double r169871 = sqrt(r169855);
        double r169872 = r169854 * r169871;
        double r169873 = r169853 ? r169870 : r169872;
        return r169873;
}

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) < -4.6771792379772145e+305

    1. Initial program 62.9

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

    3. Applied div-inv62.9

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

    4. Applied associate-*r*27.5

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

    6. Applied sqr-pow27.5

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

    7. Applied associate-*l*24.0

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

    9. Applied clear-num24.0

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

    if -4.6771792379772145e+305 < (/ h l) < -1.5075201281768343e-281

    1. Initial program 15.0

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

    3. Applied sqr-pow15.0

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

    4. Applied associate-*l*13.0

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

    if -1.5075201281768343e-281 < (/ h l)

    1. Initial program 8.0

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

    2. Taylor expanded around 0 2.9

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.677179237977214472415692974466936692272 \cdot 10^{305} \lor \frac{h}{\ell} \le \frac{-77024497644807}{5.109351192408882717840314145374306729133 \cdot 10^{294}}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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))))))