Average Error: 14.2 → 9.4
Time: 16.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \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)}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r194684 = w0;
        double r194685 = 1.0;
        double r194686 = M;
        double r194687 = D;
        double r194688 = r194686 * r194687;
        double r194689 = 2.0;
        double r194690 = d;
        double r194691 = r194689 * r194690;
        double r194692 = r194688 / r194691;
        double r194693 = pow(r194692, r194689);
        double r194694 = h;
        double r194695 = l;
        double r194696 = r194694 / r194695;
        double r194697 = r194693 * r194696;
        double r194698 = r194685 - r194697;
        double r194699 = sqrt(r194698);
        double r194700 = r194684 * r194699;
        return r194700;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r194701 = w0;
        double r194702 = 1.0;
        double r194703 = 1.0;
        double r194704 = 2.0;
        double r194705 = d;
        double r194706 = r194704 * r194705;
        double r194707 = M;
        double r194708 = D;
        double r194709 = r194707 * r194708;
        double r194710 = r194706 / r194709;
        double r194711 = r194703 / r194710;
        double r194712 = 2.0;
        double r194713 = r194704 / r194712;
        double r194714 = pow(r194711, r194713);
        double r194715 = r194709 / r194706;
        double r194716 = pow(r194715, r194713);
        double r194717 = h;
        double r194718 = r194716 * r194717;
        double r194719 = r194714 * r194718;
        double r194720 = l;
        double r194721 = r194719 / r194720;
        double r194722 = r194702 - r194721;
        double r194723 = sqrt(r194722);
        double r194724 = r194701 * r194723;
        return r194724;
}

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. Using strategy rm

  3. Applied associate-*r/10.8

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

  5. Applied sqr-pow10.8

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

  6. Applied associate-*l*9.4

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

  8. Applied clear-num9.4

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

  9. Final simplification9.4

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

Reproduce

herbie shell --seed 2020042 
(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))))))