Average Error: 14.3 → 8.7
Time: 10.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}}{h}}}\]
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)}}{\frac{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}}{h}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r273784 = w0;
        double r273785 = 1.0;
        double r273786 = M;
        double r273787 = D;
        double r273788 = r273786 * r273787;
        double r273789 = 2.0;
        double r273790 = d;
        double r273791 = r273789 * r273790;
        double r273792 = r273788 / r273791;
        double r273793 = pow(r273792, r273789);
        double r273794 = h;
        double r273795 = l;
        double r273796 = r273794 / r273795;
        double r273797 = r273793 * r273796;
        double r273798 = r273785 - r273797;
        double r273799 = sqrt(r273798);
        double r273800 = r273784 * r273799;
        return r273800;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r273801 = w0;
        double r273802 = 1.0;
        double r273803 = M;
        double r273804 = D;
        double r273805 = r273803 * r273804;
        double r273806 = 2.0;
        double r273807 = d;
        double r273808 = r273806 * r273807;
        double r273809 = r273805 / r273808;
        double r273810 = 2.0;
        double r273811 = r273806 / r273810;
        double r273812 = pow(r273809, r273811);
        double r273813 = l;
        double r273814 = r273813 / r273812;
        double r273815 = h;
        double r273816 = r273814 / r273815;
        double r273817 = r273812 / r273816;
        double r273818 = r273802 - r273817;
        double r273819 = sqrt(r273818);
        double r273820 = r273801 * r273819;
        return r273820;
}

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 14.3

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

  3. Applied associate-*r/10.7

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

  5. Applied sqr-pow10.7

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

  6. Applied associate-*l*9.1

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

  8. Applied associate-/l*8.6

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

  10. Applied associate-/r*8.7

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

  11. Final simplification8.7

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

Reproduce

herbie shell --seed 2020064 
(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))))))