Average Error: 14.1 → 8.5
Time: 28.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \left(\left(h \cdot \frac{\frac{\frac{M \cdot D}{d}}{\sqrt{2}}}{\ell}\right) \cdot \frac{1}{\sqrt{2}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \left(\left(h \cdot \frac{\frac{\frac{M \cdot D}{d}}{\sqrt{2}}}{\ell}\right) \cdot \frac{1}{\sqrt{2}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r3398880 = w0;
        double r3398881 = 1.0;
        double r3398882 = M;
        double r3398883 = D;
        double r3398884 = r3398882 * r3398883;
        double r3398885 = 2.0;
        double r3398886 = d;
        double r3398887 = r3398885 * r3398886;
        double r3398888 = r3398884 / r3398887;
        double r3398889 = pow(r3398888, r3398885);
        double r3398890 = h;
        double r3398891 = l;
        double r3398892 = r3398890 / r3398891;
        double r3398893 = r3398889 * r3398892;
        double r3398894 = r3398881 - r3398893;
        double r3398895 = sqrt(r3398894);
        double r3398896 = r3398880 * r3398895;
        return r3398896;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r3398897 = 1.0;
        double r3398898 = h;
        double r3398899 = M;
        double r3398900 = D;
        double r3398901 = r3398899 * r3398900;
        double r3398902 = d;
        double r3398903 = r3398901 / r3398902;
        double r3398904 = 2.0;
        double r3398905 = sqrt(r3398904);
        double r3398906 = r3398903 / r3398905;
        double r3398907 = l;
        double r3398908 = r3398906 / r3398907;
        double r3398909 = r3398898 * r3398908;
        double r3398910 = r3398897 / r3398905;
        double r3398911 = r3398909 * r3398910;
        double r3398912 = r3398903 / r3398904;
        double r3398913 = r3398911 * r3398912;
        double r3398914 = r3398897 - r3398913;
        double r3398915 = sqrt(r3398914);
        double r3398916 = w0;
        double r3398917 = r3398915 * r3398916;
        return r3398917;
}

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. Simplified12.2

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

  4. Applied associate-/r/8.4

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

  6. Applied *-un-lft-identity8.4

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

  7. Applied add-sqr-sqrt8.5

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

  8. Applied *-un-lft-identity8.5

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

  9. Applied times-frac8.5

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

  10. Applied times-frac8.5

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

  11. Applied associate-*l*8.5

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

  12. Final simplification8.5

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))