Average Error: 14.5 → 9.6
Time: 10.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h} \cdot \sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}\right) \cdot \sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}\right)}{\ell}}\]
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{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h} \cdot \sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}\right) \cdot \sqrt[3]{{\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 r273584 = w0;
        double r273585 = 1.0;
        double r273586 = M;
        double r273587 = D;
        double r273588 = r273586 * r273587;
        double r273589 = 2.0;
        double r273590 = d;
        double r273591 = r273589 * r273590;
        double r273592 = r273588 / r273591;
        double r273593 = pow(r273592, r273589);
        double r273594 = h;
        double r273595 = l;
        double r273596 = r273594 / r273595;
        double r273597 = r273593 * r273596;
        double r273598 = r273585 - r273597;
        double r273599 = sqrt(r273598);
        double r273600 = r273584 * r273599;
        return r273600;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r273601 = w0;
        double r273602 = 1.0;
        double r273603 = M;
        double r273604 = D;
        double r273605 = r273603 * r273604;
        double r273606 = 2.0;
        double r273607 = d;
        double r273608 = r273606 * r273607;
        double r273609 = r273605 / r273608;
        double r273610 = 2.0;
        double r273611 = r273606 / r273610;
        double r273612 = pow(r273609, r273611);
        double r273613 = h;
        double r273614 = r273612 * r273613;
        double r273615 = cbrt(r273614);
        double r273616 = r273615 * r273615;
        double r273617 = r273616 * r273615;
        double r273618 = r273612 * r273617;
        double r273619 = l;
        double r273620 = r273618 / r273619;
        double r273621 = r273602 - r273620;
        double r273622 = sqrt(r273621);
        double r273623 = r273601 * r273622;
        return r273623;
}

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

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

  9. Final simplification9.6

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

Reproduce

herbie shell --seed 2020060 
(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))))))