Average Error: 13.7 → 7.9
Time: 9.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt[3]{h}\right) \cdot \frac{1}{\ell}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt[3]{h}\right) \cdot \frac{1}{\ell}\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r260708 = w0;
        double r260709 = 1.0;
        double r260710 = M;
        double r260711 = D;
        double r260712 = r260710 * r260711;
        double r260713 = 2.0;
        double r260714 = d;
        double r260715 = r260713 * r260714;
        double r260716 = r260712 / r260715;
        double r260717 = pow(r260716, r260713);
        double r260718 = h;
        double r260719 = l;
        double r260720 = r260718 / r260719;
        double r260721 = r260717 * r260720;
        double r260722 = r260709 - r260721;
        double r260723 = sqrt(r260722);
        double r260724 = r260708 * r260723;
        return r260724;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r260725 = w0;
        double r260726 = 1.0;
        double r260727 = h;
        double r260728 = cbrt(r260727);
        double r260729 = M;
        double r260730 = D;
        double r260731 = r260729 * r260730;
        double r260732 = 2.0;
        double r260733 = d;
        double r260734 = r260732 * r260733;
        double r260735 = r260731 / r260734;
        double r260736 = 2.0;
        double r260737 = r260732 / r260736;
        double r260738 = pow(r260735, r260737);
        double r260739 = r260728 * r260738;
        double r260740 = r260739 * r260728;
        double r260741 = 1.0;
        double r260742 = l;
        double r260743 = r260741 / r260742;
        double r260744 = r260740 * r260743;
        double r260745 = r260739 * r260744;
        double r260746 = r260726 - r260745;
        double r260747 = sqrt(r260746);
        double r260748 = r260725 * r260747;
        return r260748;
}

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 13.7

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

  3. Applied *-un-lft-identity13.7

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

  4. Applied add-cube-cbrt13.8

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

  5. Applied times-frac13.8

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

  6. Applied associate-*r*11.4

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

  7. Simplified11.4

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

  9. Applied sqr-pow11.4

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

  10. Applied unswap-sqr10.3

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

  12. Applied associate-*l*9.1

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

  14. Applied div-inv9.1

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

  15. Applied associate-*r*7.9

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

  16. Final simplification7.9

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

Reproduce

herbie shell --seed 2020057 
(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))))))