Average Error: 14.5 → 9.6
Time: 9.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) \cdot \sqrt[3]{h}\right)}{\ell}}\]
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{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 \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) \cdot \sqrt[3]{h}\right)}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r163624 = w0;
        double r163625 = 1.0;
        double r163626 = M;
        double r163627 = D;
        double r163628 = r163626 * r163627;
        double r163629 = 2.0;
        double r163630 = d;
        double r163631 = r163629 * r163630;
        double r163632 = r163628 / r163631;
        double r163633 = pow(r163632, r163629);
        double r163634 = h;
        double r163635 = l;
        double r163636 = r163634 / r163635;
        double r163637 = r163633 * r163636;
        double r163638 = r163625 - r163637;
        double r163639 = sqrt(r163638);
        double r163640 = r163624 * r163639;
        return r163640;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r163641 = w0;
        double r163642 = 1.0;
        double r163643 = M;
        double r163644 = D;
        double r163645 = r163643 * r163644;
        double r163646 = 2.0;
        double r163647 = d;
        double r163648 = r163646 * r163647;
        double r163649 = r163645 / r163648;
        double r163650 = 2.0;
        double r163651 = r163646 / r163650;
        double r163652 = pow(r163649, r163651);
        double r163653 = h;
        double r163654 = cbrt(r163653);
        double r163655 = r163654 * r163654;
        double r163656 = r163652 * r163655;
        double r163657 = r163656 * r163654;
        double r163658 = r163652 * r163657;
        double r163659 = l;
        double r163660 = r163658 / r163659;
        double r163661 = r163642 - r163660;
        double r163662 = sqrt(r163661);
        double r163663 = r163641 * r163662;
        return r163663;
}

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 14.5

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

  3. Applied associate-*r/11.0

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

  5. Applied sqr-pow11.0

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

  6. Applied associate-*l*9.6

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

  8. Applied add-cube-cbrt9.6

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

  9. Applied associate-*r*9.6

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

  10. Final simplification9.6

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

Reproduce

herbie shell --seed 2020060 +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))))))