Average Error: 14.2 → 9.4
Time: 17.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\]
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(\frac{1}{\frac{2 \cdot d}{M \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 r263693 = w0;
        double r263694 = 1.0;
        double r263695 = M;
        double r263696 = D;
        double r263697 = r263695 * r263696;
        double r263698 = 2.0;
        double r263699 = d;
        double r263700 = r263698 * r263699;
        double r263701 = r263697 / r263700;
        double r263702 = pow(r263701, r263698);
        double r263703 = h;
        double r263704 = l;
        double r263705 = r263703 / r263704;
        double r263706 = r263702 * r263705;
        double r263707 = r263694 - r263706;
        double r263708 = sqrt(r263707);
        double r263709 = r263693 * r263708;
        return r263709;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r263710 = w0;
        double r263711 = 1.0;
        double r263712 = M;
        double r263713 = D;
        double r263714 = r263712 * r263713;
        double r263715 = 2.0;
        double r263716 = d;
        double r263717 = r263715 * r263716;
        double r263718 = r263714 / r263717;
        double r263719 = 2.0;
        double r263720 = r263715 / r263719;
        double r263721 = pow(r263718, r263720);
        double r263722 = 1.0;
        double r263723 = r263717 / r263714;
        double r263724 = r263722 / r263723;
        double r263725 = pow(r263724, r263720);
        double r263726 = h;
        double r263727 = r263725 * r263726;
        double r263728 = r263721 * r263727;
        double r263729 = l;
        double r263730 = r263728 / r263729;
        double r263731 = r263711 - r263730;
        double r263732 = sqrt(r263731);
        double r263733 = r263710 * r263732;
        return r263733;
}

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 14.2

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

  3. Applied associate-*r/10.8

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

  5. Applied sqr-pow10.8

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

  6. Applied associate-*l*9.4

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

  8. Applied clear-num9.4

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

  9. Final simplification9.4

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

Reproduce

herbie shell --seed 2020042 
(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))))))