Average Error: 14.4 → 8.8
Time: 23.1s
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 r178801 = w0;
        double r178802 = 1.0;
        double r178803 = M;
        double r178804 = D;
        double r178805 = r178803 * r178804;
        double r178806 = 2.0;
        double r178807 = d;
        double r178808 = r178806 * r178807;
        double r178809 = r178805 / r178808;
        double r178810 = pow(r178809, r178806);
        double r178811 = h;
        double r178812 = l;
        double r178813 = r178811 / r178812;
        double r178814 = r178810 * r178813;
        double r178815 = r178802 - r178814;
        double r178816 = sqrt(r178815);
        double r178817 = r178801 * r178816;
        return r178817;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r178818 = w0;
        double r178819 = 1.0;
        double r178820 = 1.0;
        double r178821 = 2.0;
        double r178822 = d;
        double r178823 = r178821 * r178822;
        double r178824 = M;
        double r178825 = D;
        double r178826 = r178824 * r178825;
        double r178827 = r178823 / r178826;
        double r178828 = r178820 / r178827;
        double r178829 = 2.0;
        double r178830 = r178821 / r178829;
        double r178831 = pow(r178828, r178830);
        double r178832 = r178820 / r178823;
        double r178833 = r178826 * r178832;
        double r178834 = pow(r178833, r178830);
        double r178835 = h;
        double r178836 = r178834 * r178835;
        double r178837 = l;
        double r178838 = r178820 / r178837;
        double r178839 = r178836 * r178838;
        double r178840 = r178831 * r178839;
        double r178841 = r178819 - r178840;
        double r178842 = sqrt(r178841);
        double r178843 = r178818 * r178842;
        return r178843;
}

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))))))