Average Error: 13.6 → 9.2
Time: 29.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.459510325973658786992790942635709262941 \cdot 10^{-131} \lor \neg \left(\frac{h}{\ell} \le -0.0\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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 -2.459510325973658786992790942635709262941 \cdot 10^{-131} \lor \neg \left(\frac{h}{\ell} \le -0.0\right):\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;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)}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r146810 = w0;
        double r146811 = 1.0;
        double r146812 = M;
        double r146813 = D;
        double r146814 = r146812 * r146813;
        double r146815 = 2.0;
        double r146816 = d;
        double r146817 = r146815 * r146816;
        double r146818 = r146814 / r146817;
        double r146819 = pow(r146818, r146815);
        double r146820 = h;
        double r146821 = l;
        double r146822 = r146820 / r146821;
        double r146823 = r146819 * r146822;
        double r146824 = r146811 - r146823;
        double r146825 = sqrt(r146824);
        double r146826 = r146810 * r146825;
        return r146826;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r146827 = h;
        double r146828 = l;
        double r146829 = r146827 / r146828;
        double r146830 = -2.459510325973659e-131;
        bool r146831 = r146829 <= r146830;
        double r146832 = -0.0;
        bool r146833 = r146829 <= r146832;
        double r146834 = !r146833;
        bool r146835 = r146831 || r146834;
        double r146836 = w0;
        double r146837 = 1.0;
        double r146838 = M;
        double r146839 = 2.0;
        double r146840 = d;
        double r146841 = r146839 * r146840;
        double r146842 = D;
        double r146843 = r146841 / r146842;
        double r146844 = r146838 / r146843;
        double r146845 = pow(r146844, r146839);
        double r146846 = r146845 * r146827;
        double r146847 = 1.0;
        double r146848 = r146847 / r146828;
        double r146849 = r146846 * r146848;
        double r146850 = r146837 - r146849;
        double r146851 = sqrt(r146850);
        double r146852 = r146836 * r146851;
        double r146853 = r146838 * r146842;
        double r146854 = r146853 / r146841;
        double r146855 = 2.0;
        double r146856 = r146839 / r146855;
        double r146857 = pow(r146854, r146856);
        double r146858 = r146857 * r146829;
        double r146859 = r146857 * r146858;
        double r146860 = r146837 - r146859;
        double r146861 = sqrt(r146860);
        double r146862 = r146836 * r146861;
        double r146863 = r146835 ? r146852 : r146862;
        return r146863;
}

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 (/ h l) < -2.459510325973659e-131 or -0.0 < (/ h l)

    1. Initial program 13.0

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

    3. Applied div-inv13.0

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

    4. Applied associate-*r*9.2

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

    6. Applied associate-/l*9.1

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

    if -2.459510325973659e-131 < (/ h l) < -0.0

    1. Initial program 15.7

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

    3. Applied sqr-pow15.7

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

    4. Applied associate-*l*11.0

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.459510325973658786992790942635709262941 \cdot 10^{-131} \lor \neg \left(\frac{h}{\ell} \le -0.0\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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))))))