Average Error: 14.0 → 9.3
Time: 10.9s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \left(\left|\sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\right| \cdot \sqrt{\sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}}\right)\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \left(\left|\sqrt[3]{1 - \frac{{\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}}\right| \cdot \sqrt{\sqrt[3]{1 - \frac{{\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}}}\right)
double f(double w0, double M, double D, double h, double l, double d) {
        double r187584 = w0;
        double r187585 = 1.0;
        double r187586 = M;
        double r187587 = D;
        double r187588 = r187586 * r187587;
        double r187589 = 2.0;
        double r187590 = d;
        double r187591 = r187589 * r187590;
        double r187592 = r187588 / r187591;
        double r187593 = pow(r187592, r187589);
        double r187594 = h;
        double r187595 = l;
        double r187596 = r187594 / r187595;
        double r187597 = r187593 * r187596;
        double r187598 = r187585 - r187597;
        double r187599 = sqrt(r187598);
        double r187600 = r187584 * r187599;
        return r187600;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r187601 = w0;
        double r187602 = 1.0;
        double r187603 = M;
        double r187604 = D;
        double r187605 = r187603 * r187604;
        double r187606 = 2.0;
        double r187607 = d;
        double r187608 = r187606 * r187607;
        double r187609 = r187605 / r187608;
        double r187610 = 2.0;
        double r187611 = r187606 / r187610;
        double r187612 = pow(r187609, r187611);
        double r187613 = h;
        double r187614 = r187612 * r187613;
        double r187615 = r187612 * r187614;
        double r187616 = l;
        double r187617 = r187615 / r187616;
        double r187618 = r187602 - r187617;
        double r187619 = cbrt(r187618);
        double r187620 = fabs(r187619);
        double r187621 = sqrt(r187619);
        double r187622 = r187620 * r187621;
        double r187623 = r187601 * r187622;
        return r187623;
}

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 14.0

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

  3. Applied associate-*r/10.6

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

  5. Applied sqr-pow10.6

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

  6. Applied associate-*l*9.3

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

  8. Applied add-cube-cbrt9.3

    \[\leadsto w0 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}} \cdot \sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\right) \cdot \sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}}}\]

  9. Applied sqrt-prod9.3

    \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}} \cdot \sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}} \cdot \sqrt{\sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}}\right)}\]

  10. Simplified9.3

    \[\leadsto w0 \cdot \left(\color{blue}{\left|\sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\right|} \cdot \sqrt{\sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}}\right)\]

  11. Final simplification9.3

    \[\leadsto w0 \cdot \left(\left|\sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\right| \cdot \sqrt{\sqrt[3]{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}}\right)\]

Reproduce

herbie shell --seed 2020062 
(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))))))