Average Error: 14.4 → 8.8
Time: 23.4s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r177833 = w0;
        double r177834 = 1.0;
        double r177835 = M;
        double r177836 = D;
        double r177837 = r177835 * r177836;
        double r177838 = 2.0;
        double r177839 = d;
        double r177840 = r177838 * r177839;
        double r177841 = r177837 / r177840;
        double r177842 = pow(r177841, r177838);
        double r177843 = h;
        double r177844 = l;
        double r177845 = r177843 / r177844;
        double r177846 = r177842 * r177845;
        double r177847 = r177834 - r177846;
        double r177848 = sqrt(r177847);
        double r177849 = r177833 * r177848;
        return r177849;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r177850 = w0;
        double r177851 = 1.0;
        double r177852 = 1.0;
        double r177853 = 2.0;
        double r177854 = d;
        double r177855 = r177853 * r177854;
        double r177856 = M;
        double r177857 = D;
        double r177858 = r177856 * r177857;
        double r177859 = r177855 / r177858;
        double r177860 = r177852 / r177859;
        double r177861 = 2.0;
        double r177862 = r177853 / r177861;
        double r177863 = pow(r177860, r177862);
        double r177864 = r177852 / r177855;
        double r177865 = r177858 * r177864;
        double r177866 = pow(r177865, r177862);
        double r177867 = h;
        double r177868 = r177866 * r177867;
        double r177869 = l;
        double r177870 = r177852 / r177869;
        double r177871 = r177868 * r177870;
        double r177872 = r177863 * r177871;
        double r177873 = r177851 - r177872;
        double r177874 = sqrt(r177873);
        double r177875 = r177850 * r177874;
        return r177875;
}

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 14.4

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

  3. Applied sqr-pow14.4

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

  4. Applied associate-*l*12.6

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

  6. Applied div-inv12.6

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

  7. Applied associate-*r*8.8

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

  9. Applied clear-num8.8

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

  11. Applied div-inv8.8

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

  12. Final simplification8.8

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

Reproduce

herbie shell --seed 2019325 
(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))))))