Average Error: 13.9 → 9.2
Time: 22.4s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\]
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 r125601 = w0;
        double r125602 = 1.0;
        double r125603 = M;
        double r125604 = D;
        double r125605 = r125603 * r125604;
        double r125606 = 2.0;
        double r125607 = d;
        double r125608 = r125606 * r125607;
        double r125609 = r125605 / r125608;
        double r125610 = pow(r125609, r125606);
        double r125611 = h;
        double r125612 = l;
        double r125613 = r125611 / r125612;
        double r125614 = r125610 * r125613;
        double r125615 = r125602 - r125614;
        double r125616 = sqrt(r125615);
        double r125617 = r125601 * r125616;
        return r125617;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r125618 = w0;
        double r125619 = 1.0;
        double r125620 = 1.0;
        double r125621 = 2.0;
        double r125622 = d;
        double r125623 = r125621 * r125622;
        double r125624 = M;
        double r125625 = D;
        double r125626 = r125624 * r125625;
        double r125627 = r125623 / r125626;
        double r125628 = r125620 / r125627;
        double r125629 = 2.0;
        double r125630 = r125621 / r125629;
        double r125631 = pow(r125628, r125630);
        double r125632 = r125626 / r125623;
        double r125633 = pow(r125632, r125630);
        double r125634 = h;
        double r125635 = r125633 * r125634;
        double r125636 = r125631 * r125635;
        double r125637 = l;
        double r125638 = r125636 / r125637;
        double r125639 = r125619 - r125638;
        double r125640 = sqrt(r125639);
        double r125641 = r125618 * r125640;
        return r125641;
}

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 13.9

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

  3. Applied associate-*r/10.7

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

  5. Applied sqr-pow10.7

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

  6. Applied associate-*l*9.2

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

  8. Applied clear-num9.2

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

  9. Final simplification9.2

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

Reproduce

herbie shell --seed 2019235 
(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))))))