Average Error: 14.1 → 8.7
Time: 32.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{D \cdot M}{2} \cdot \frac{1}{d}\right)}^{\left(\frac{2}{2}\right)}}{\ell} \cdot \left({\left(\frac{\frac{M}{\frac{2}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}\]
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{D \cdot M}{2} \cdot \frac{1}{d}\right)}^{\left(\frac{2}{2}\right)}}{\ell} \cdot \left({\left(\frac{\frac{M}{\frac{2}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r169026 = w0;
        double r169027 = 1.0;
        double r169028 = M;
        double r169029 = D;
        double r169030 = r169028 * r169029;
        double r169031 = 2.0;
        double r169032 = d;
        double r169033 = r169031 * r169032;
        double r169034 = r169030 / r169033;
        double r169035 = pow(r169034, r169031);
        double r169036 = h;
        double r169037 = l;
        double r169038 = r169036 / r169037;
        double r169039 = r169035 * r169038;
        double r169040 = r169027 - r169039;
        double r169041 = sqrt(r169040);
        double r169042 = r169026 * r169041;
        return r169042;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r169043 = w0;
        double r169044 = 1.0;
        double r169045 = D;
        double r169046 = M;
        double r169047 = r169045 * r169046;
        double r169048 = 2.0;
        double r169049 = r169047 / r169048;
        double r169050 = 1.0;
        double r169051 = d;
        double r169052 = r169050 / r169051;
        double r169053 = r169049 * r169052;
        double r169054 = 2.0;
        double r169055 = r169048 / r169054;
        double r169056 = pow(r169053, r169055);
        double r169057 = l;
        double r169058 = r169056 / r169057;
        double r169059 = r169048 / r169045;
        double r169060 = r169046 / r169059;
        double r169061 = r169060 / r169051;
        double r169062 = pow(r169061, r169055);
        double r169063 = h;
        double r169064 = r169062 * r169063;
        double r169065 = r169058 * r169064;
        double r169066 = r169044 - r169065;
        double r169067 = sqrt(r169066);
        double r169068 = r169043 * r169067;
        return r169068;
}

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 14.1

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

  2. Simplified11.0

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

  4. Applied sqr-pow11.0

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

  5. Applied associate-*r*9.4

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

  6. Simplified9.4

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

  8. Applied div-inv9.4

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

  10. Applied associate-*l*8.8

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

  11. Simplified8.7

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

  13. Applied div-inv8.7

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

  14. Final simplification8.7

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

Reproduce

herbie shell --seed 2019194 
(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))))))