Average Error: 13.7 → 8.1
Time: 2.4m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \left(\sqrt{\sqrt[3]{1 - \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}} \cdot \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\frac{2}{M} \cdot \frac{d}{D}}} \cdot \sqrt[3]{1 - \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}} \cdot \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\frac{2}{M} \cdot \frac{d}{D}}}} \cdot \sqrt{\sqrt[3]{1 - \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}} \cdot \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\frac{2}{M} \cdot \frac{d}{D}}}}\right)\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \left(\sqrt{\sqrt[3]{1 - \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}} \cdot \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\frac{2}{M} \cdot \frac{d}{D}}} \cdot \sqrt[3]{1 - \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}} \cdot \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\frac{2}{M} \cdot \frac{d}{D}}}} \cdot \sqrt{\sqrt[3]{1 - \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}} \cdot \frac{\frac{\sqrt[3]{h}}{\frac{\sqrt[3]{\frac{2}{M} \cdot \frac{d}{D}}}{\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}}}{\frac{2}{M} \cdot \frac{d}{D}}}}\right)
double f(double w0, double M, double D, double h, double l, double d) {
        double r8881817 = w0;
        double r8881818 = 1.0;
        double r8881819 = M;
        double r8881820 = D;
        double r8881821 = r8881819 * r8881820;
        double r8881822 = 2.0;
        double r8881823 = d;
        double r8881824 = r8881822 * r8881823;
        double r8881825 = r8881821 / r8881824;
        double r8881826 = pow(r8881825, r8881822);
        double r8881827 = h;
        double r8881828 = l;
        double r8881829 = r8881827 / r8881828;
        double r8881830 = r8881826 * r8881829;
        double r8881831 = r8881818 - r8881830;
        double r8881832 = sqrt(r8881831);
        double r8881833 = r8881817 * r8881832;
        return r8881833;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r8881834 = w0;
        double r8881835 = 1.0;
        double r8881836 = h;
        double r8881837 = cbrt(r8881836);
        double r8881838 = 2.0;
        double r8881839 = M;
        double r8881840 = r8881838 / r8881839;
        double r8881841 = d;
        double r8881842 = D;
        double r8881843 = r8881841 / r8881842;
        double r8881844 = r8881840 * r8881843;
        double r8881845 = cbrt(r8881844);
        double r8881846 = l;
        double r8881847 = cbrt(r8881846);
        double r8881848 = r8881837 / r8881847;
        double r8881849 = cbrt(r8881848);
        double r8881850 = r8881845 / r8881849;
        double r8881851 = r8881837 / r8881850;
        double r8881852 = r8881847 * r8881847;
        double r8881853 = r8881852 * r8881850;
        double r8881854 = r8881851 / r8881853;
        double r8881855 = r8881851 / r8881844;
        double r8881856 = r8881854 * r8881855;
        double r8881857 = r8881835 - r8881856;
        double r8881858 = cbrt(r8881857);
        double r8881859 = r8881858 * r8881858;
        double r8881860 = sqrt(r8881859);
        double r8881861 = sqrt(r8881858);
        double r8881862 = r8881860 * r8881861;
        double r8881863 = r8881834 * r8881862;
        return r8881863;
}

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. Initial program 13.7

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

  2. Simplified12.3

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

  4. Applied add-cube-cbrt12.3

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

  5. Applied add-cube-cbrt12.3

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

  6. Applied times-frac12.3

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

  7. Applied associate-/l*8.8

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

  9. Applied add-cube-cbrt8.8

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

  10. Applied add-cube-cbrt8.8

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

  11. Applied times-frac8.8

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

  12. Applied div-inv8.8

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

  13. Applied times-frac8.2

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

  15. Applied *-un-lft-identity8.2

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

  16. Applied sqrt-prod8.2

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

  17. Simplified8.2

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

  18. Simplified8.0

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

  20. Applied add-cube-cbrt8.1

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

  21. Applied sqrt-prod8.1

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

  22. Final simplification8.1

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

Reproduce

herbie shell --seed 2019163 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))