Average Error: 13.7 → 8.1
Time: 27.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}} \cdot \sqrt[3]{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\right) \cdot \sqrt[3]{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}} \cdot \sqrt[3]{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\right) \cdot \sqrt[3]{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r120674 = w0;
        double r120675 = 1.0;
        double r120676 = M;
        double r120677 = D;
        double r120678 = r120676 * r120677;
        double r120679 = 2.0;
        double r120680 = d;
        double r120681 = r120679 * r120680;
        double r120682 = r120678 / r120681;
        double r120683 = pow(r120682, r120679);
        double r120684 = h;
        double r120685 = l;
        double r120686 = r120684 / r120685;
        double r120687 = r120683 * r120686;
        double r120688 = r120675 - r120687;
        double r120689 = sqrt(r120688);
        double r120690 = r120674 * r120689;
        return r120690;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r120691 = w0;
        double r120692 = 1.0;
        double r120693 = M;
        double r120694 = D;
        double r120695 = r120693 * r120694;
        double r120696 = 2.0;
        double r120697 = d;
        double r120698 = r120696 * r120697;
        double r120699 = r120695 / r120698;
        double r120700 = 2.0;
        double r120701 = r120696 / r120700;
        double r120702 = pow(r120699, r120701);
        double r120703 = h;
        double r120704 = r120702 * r120703;
        double r120705 = l;
        double r120706 = r120704 / r120705;
        double r120707 = cbrt(r120706);
        double r120708 = r120707 * r120707;
        double r120709 = r120708 * r120707;
        double r120710 = r120702 * r120709;
        double r120711 = r120692 - r120710;
        double r120712 = sqrt(r120711);
        double r120713 = r120691 * r120712;
        return r120713;
}

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 associate-*r/10.5

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

  5. Applied sqr-pow10.5

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

  6. Applied associate-*l*8.9

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

  8. Applied *-un-lft-identity8.9

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

  9. Applied times-frac8.1

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

  10. Simplified8.1

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

  12. Applied add-cube-cbrt8.1

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

  13. Final simplification8.1

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

Reproduce

herbie shell --seed 2019195 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))