Average Error: 14.2 → 7.9
Time: 34.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\frac{1}{2} \cdot M}{\frac{d}{D}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\frac{1}{2} \cdot M}{\frac{d}{D}}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\frac{1}{2} \cdot M}{\frac{d}{D}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\frac{1}{2} \cdot M}{\frac{d}{D}}\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r5398803 = w0;
        double r5398804 = 1.0;
        double r5398805 = M;
        double r5398806 = D;
        double r5398807 = r5398805 * r5398806;
        double r5398808 = 2.0;
        double r5398809 = d;
        double r5398810 = r5398808 * r5398809;
        double r5398811 = r5398807 / r5398810;
        double r5398812 = pow(r5398811, r5398808);
        double r5398813 = h;
        double r5398814 = l;
        double r5398815 = r5398813 / r5398814;
        double r5398816 = r5398812 * r5398815;
        double r5398817 = r5398804 - r5398816;
        double r5398818 = sqrt(r5398817);
        double r5398819 = r5398803 * r5398818;
        return r5398819;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5398820 = w0;
        double r5398821 = 1.0;
        double r5398822 = h;
        double r5398823 = cbrt(r5398822);
        double r5398824 = l;
        double r5398825 = cbrt(r5398824);
        double r5398826 = r5398823 / r5398825;
        double r5398827 = 0.5;
        double r5398828 = M;
        double r5398829 = r5398827 * r5398828;
        double r5398830 = d;
        double r5398831 = D;
        double r5398832 = r5398830 / r5398831;
        double r5398833 = r5398829 / r5398832;
        double r5398834 = r5398826 * r5398833;
        double r5398835 = r5398834 * r5398826;
        double r5398836 = r5398835 * r5398834;
        double r5398837 = r5398821 - r5398836;
        double r5398838 = sqrt(r5398837);
        double r5398839 = r5398820 * r5398838;
        return r5398839;
}

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 14.2

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

  2. Simplified14.2

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

  4. Applied add-cube-cbrt14.2

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

  5. Applied add-cube-cbrt14.3

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

  6. Applied times-frac14.3

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

  7. Applied associate-*r*11.0

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

  8. Simplified8.2

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

  10. Applied add-cbrt-cube9.6

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

  11. Simplified9.6

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

  13. Applied add-sqr-sqrt9.6

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

  14. Applied pow39.6

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

  15. Applied rem-cbrt-cube7.9

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

  16. Final simplification7.9

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

Reproduce

herbie shell --seed 2019142 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))