Average Error: 14.1 → 9.6
Time: 24.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -3.3635527713506517 \cdot 10^{+86}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h} \cdot \left(\left(\frac{M \cdot D}{2} \cdot \sqrt[3]{h}\right) \cdot \left(\frac{\frac{M \cdot D}{2}}{d} \cdot \sqrt[3]{h}\right)\right)}{d \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt[3]{\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}}\right)} \cdot \left(1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}}\right)\right)}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -3.3635527713506517 \cdot 10^{+86}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h} \cdot \left(\left(\frac{M \cdot D}{2} \cdot \sqrt[3]{h}\right) \cdot \left(\frac{\frac{M \cdot D}{2}}{d} \cdot \sqrt[3]{h}\right)\right)}{d \cdot \ell}}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt[3]{\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}}\right)} \cdot \left(1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}}\right)\right)}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r2336798 = w0;
        double r2336799 = 1.0;
        double r2336800 = M;
        double r2336801 = D;
        double r2336802 = r2336800 * r2336801;
        double r2336803 = 2.0;
        double r2336804 = d;
        double r2336805 = r2336803 * r2336804;
        double r2336806 = r2336802 / r2336805;
        double r2336807 = pow(r2336806, r2336803);
        double r2336808 = h;
        double r2336809 = l;
        double r2336810 = r2336808 / r2336809;
        double r2336811 = r2336807 * r2336810;
        double r2336812 = r2336799 - r2336811;
        double r2336813 = sqrt(r2336812);
        double r2336814 = r2336798 * r2336813;
        return r2336814;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r2336815 = M;
        double r2336816 = D;
        double r2336817 = r2336815 * r2336816;
        double r2336818 = 2.0;
        double r2336819 = d;
        double r2336820 = r2336818 * r2336819;
        double r2336821 = r2336817 / r2336820;
        double r2336822 = -3.3635527713506517e+86;
        bool r2336823 = r2336821 <= r2336822;
        double r2336824 = w0;
        double r2336825 = 1.0;
        double r2336826 = h;
        double r2336827 = cbrt(r2336826);
        double r2336828 = r2336817 / r2336818;
        double r2336829 = r2336828 * r2336827;
        double r2336830 = r2336828 / r2336819;
        double r2336831 = r2336830 * r2336827;
        double r2336832 = r2336829 * r2336831;
        double r2336833 = r2336827 * r2336832;
        double r2336834 = l;
        double r2336835 = r2336819 * r2336834;
        double r2336836 = r2336833 / r2336835;
        double r2336837 = r2336825 - r2336836;
        double r2336838 = sqrt(r2336837);
        double r2336839 = r2336824 * r2336838;
        double r2336840 = cbrt(r2336834);
        double r2336841 = r2336827 / r2336840;
        double r2336842 = r2336819 / r2336816;
        double r2336843 = r2336815 / r2336842;
        double r2336844 = r2336843 / r2336818;
        double r2336845 = r2336840 / r2336844;
        double r2336846 = r2336827 / r2336845;
        double r2336847 = r2336846 * r2336846;
        double r2336848 = r2336841 * r2336847;
        double r2336849 = r2336825 - r2336848;
        double r2336850 = sqrt(r2336849);
        double r2336851 = r2336850 * r2336849;
        double r2336852 = cbrt(r2336851);
        double r2336853 = r2336824 * r2336852;
        double r2336854 = r2336823 ? r2336839 : r2336853;
        return r2336854;
}

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 (/ (* M D) (* 2 d)) < -3.3635527713506517e+86

    1. Initial program 47.1

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

    2. Simplified47.1

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

    4. Applied *-un-lft-identity47.1

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

    5. Applied add-cube-cbrt47.1

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

    6. Applied times-frac47.1

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

    7. Applied associate-*r*48.2

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

    8. Simplified42.7

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

    10. Applied associate-*l/42.7

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

    11. Applied associate-*r/43.7

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

    12. Applied frac-times41.3

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

    if -3.3635527713506517e+86 < (/ (* M D) (* 2 d))

    1. Initial program 10.9

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

    2. Simplified10.9

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

    4. Applied *-un-lft-identity10.9

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

    5. Applied add-cube-cbrt10.9

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

    6. Applied times-frac10.9

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

    7. Applied associate-*r*7.9

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

    8. Simplified7.2

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

    10. Applied add-cube-cbrt7.2

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

    11. Applied *-un-lft-identity7.2

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

    12. Applied cbrt-prod7.2

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

    13. Applied times-frac7.2

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

    14. Applied associate-*r*6.0

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

    15. Simplified5.9

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

    17. Applied add-cbrt-cube6.9

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

    18. Simplified6.5

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -3.3635527713506517 \cdot 10^{+86}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h} \cdot \left(\left(\frac{M \cdot D}{2} \cdot \sqrt[3]{h}\right) \cdot \left(\frac{\frac{M \cdot D}{2}}{d} \cdot \sqrt[3]{h}\right)\right)}{d \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt[3]{\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}}\right)} \cdot \left(1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}} \cdot \frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\ell}}{\frac{\frac{M}{\frac{d}{D}}}{2}}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))