Average Error: 14.0 → 9.3
Time: 10.2s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\]
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(\left(M \cdot D\right) \cdot \frac{1}{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 r273783 = w0;
        double r273784 = 1.0;
        double r273785 = M;
        double r273786 = D;
        double r273787 = r273785 * r273786;
        double r273788 = 2.0;
        double r273789 = d;
        double r273790 = r273788 * r273789;
        double r273791 = r273787 / r273790;
        double r273792 = pow(r273791, r273788);
        double r273793 = h;
        double r273794 = l;
        double r273795 = r273793 / r273794;
        double r273796 = r273792 * r273795;
        double r273797 = r273784 - r273796;
        double r273798 = sqrt(r273797);
        double r273799 = r273783 * r273798;
        return r273799;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r273800 = w0;
        double r273801 = 1.0;
        double r273802 = M;
        double r273803 = D;
        double r273804 = r273802 * r273803;
        double r273805 = 2.0;
        double r273806 = d;
        double r273807 = r273805 * r273806;
        double r273808 = r273804 / r273807;
        double r273809 = 2.0;
        double r273810 = r273805 / r273809;
        double r273811 = pow(r273808, r273810);
        double r273812 = 1.0;
        double r273813 = r273812 / r273807;
        double r273814 = r273804 * r273813;
        double r273815 = pow(r273814, r273810);
        double r273816 = h;
        double r273817 = r273815 * r273816;
        double r273818 = r273811 * r273817;
        double r273819 = l;
        double r273820 = r273818 / r273819;
        double r273821 = r273801 - r273820;
        double r273822 = sqrt(r273821);
        double r273823 = r273800 * r273822;
        return r273823;
}

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 14.0

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

  3. Applied associate-*r/10.6

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

  5. Applied sqr-pow10.6

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

  6. Applied associate-*l*9.3

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

  8. Applied div-inv9.3

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

  9. Final simplification9.3

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

Reproduce

herbie shell --seed 2020062 +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))))))