Average Error: 13.5 → 8.1
Time: 27.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\frac{h}{\frac{\frac{d}{\sqrt[3]{\frac{M \cdot D}{2}} \cdot \sqrt[3]{\frac{M \cdot D}{2}}}}{\sqrt[3]{\frac{M \cdot D}{2}}} \cdot \ell}}{\frac{1}{\frac{M \cdot D}{2}} \cdot d}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \frac{\frac{h}{\frac{\frac{d}{\sqrt[3]{\frac{M \cdot D}{2}} \cdot \sqrt[3]{\frac{M \cdot D}{2}}}}{\sqrt[3]{\frac{M \cdot D}{2}}} \cdot \ell}}{\frac{1}{\frac{M \cdot D}{2}} \cdot d}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r5279823 = w0;
        double r5279824 = 1.0;
        double r5279825 = M;
        double r5279826 = D;
        double r5279827 = r5279825 * r5279826;
        double r5279828 = 2.0;
        double r5279829 = d;
        double r5279830 = r5279828 * r5279829;
        double r5279831 = r5279827 / r5279830;
        double r5279832 = pow(r5279831, r5279828);
        double r5279833 = h;
        double r5279834 = l;
        double r5279835 = r5279833 / r5279834;
        double r5279836 = r5279832 * r5279835;
        double r5279837 = r5279824 - r5279836;
        double r5279838 = sqrt(r5279837);
        double r5279839 = r5279823 * r5279838;
        return r5279839;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5279840 = 1.0;
        double r5279841 = h;
        double r5279842 = d;
        double r5279843 = M;
        double r5279844 = D;
        double r5279845 = r5279843 * r5279844;
        double r5279846 = 2.0;
        double r5279847 = r5279845 / r5279846;
        double r5279848 = cbrt(r5279847);
        double r5279849 = r5279848 * r5279848;
        double r5279850 = r5279842 / r5279849;
        double r5279851 = r5279850 / r5279848;
        double r5279852 = l;
        double r5279853 = r5279851 * r5279852;
        double r5279854 = r5279841 / r5279853;
        double r5279855 = r5279840 / r5279847;
        double r5279856 = r5279855 * r5279842;
        double r5279857 = r5279854 / r5279856;
        double r5279858 = r5279840 - r5279857;
        double r5279859 = sqrt(r5279858);
        double r5279860 = w0;
        double r5279861 = r5279859 * r5279860;
        return r5279861;
}

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 13.5

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

  2. Simplified11.8

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

  4. Applied associate-/l/8.0

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

  6. Applied add-cube-cbrt8.1

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

  7. Applied associate-/r*8.1

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

  9. Applied div-inv8.1

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

  10. Final simplification8.1

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

Reproduce

herbie shell --seed 2019158 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))