Average Error: 13.6 → 8.3
Time: 11.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}\]
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(\frac{M \cdot D}{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 r177710 = w0;
        double r177711 = 1.0;
        double r177712 = M;
        double r177713 = D;
        double r177714 = r177712 * r177713;
        double r177715 = 2.0;
        double r177716 = d;
        double r177717 = r177715 * r177716;
        double r177718 = r177714 / r177717;
        double r177719 = pow(r177718, r177715);
        double r177720 = h;
        double r177721 = l;
        double r177722 = r177720 / r177721;
        double r177723 = r177719 * r177722;
        double r177724 = r177711 - r177723;
        double r177725 = sqrt(r177724);
        double r177726 = r177710 * r177725;
        return r177726;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r177727 = w0;
        double r177728 = 1.0;
        double r177729 = 1.0;
        double r177730 = 2.0;
        double r177731 = d;
        double r177732 = r177730 * r177731;
        double r177733 = M;
        double r177734 = D;
        double r177735 = r177733 * r177734;
        double r177736 = r177732 / r177735;
        double r177737 = r177729 / r177736;
        double r177738 = 2.0;
        double r177739 = r177730 / r177738;
        double r177740 = pow(r177737, r177739);
        double r177741 = r177735 / r177732;
        double r177742 = pow(r177741, r177739);
        double r177743 = h;
        double r177744 = r177742 * r177743;
        double r177745 = l;
        double r177746 = r177729 / r177745;
        double r177747 = r177744 * r177746;
        double r177748 = r177740 * r177747;
        double r177749 = r177728 - r177748;
        double r177750 = sqrt(r177749);
        double r177751 = r177727 * r177750;
        return r177751;
}

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 13.6

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

  3. Applied div-inv13.6

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

  4. Applied associate-*r*10.3

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

  6. Applied sqr-pow10.3

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

  7. Applied associate-*l*8.8

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

  9. Applied associate-*l*8.3

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

  11. Applied clear-num8.3

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

  12. Final simplification8.3

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(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))))))