Average Error: 13.7 → 8.2
Time: 37.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\frac{\frac{D \cdot M}{2}}{d}}{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \frac{\frac{\frac{\frac{D \cdot M}{2}}{d}}{\ell}}{\frac{1}{\sqrt[3]{h}}}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \frac{\frac{\frac{D \cdot M}{2}}{d}}{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \frac{\frac{\frac{\frac{D \cdot M}{2}}{d}}{\ell}}{\frac{1}{\sqrt[3]{h}}}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r5109898 = w0;
        double r5109899 = 1.0;
        double r5109900 = M;
        double r5109901 = D;
        double r5109902 = r5109900 * r5109901;
        double r5109903 = 2.0;
        double r5109904 = d;
        double r5109905 = r5109903 * r5109904;
        double r5109906 = r5109902 / r5109905;
        double r5109907 = pow(r5109906, r5109903);
        double r5109908 = h;
        double r5109909 = l;
        double r5109910 = r5109908 / r5109909;
        double r5109911 = r5109907 * r5109910;
        double r5109912 = r5109899 - r5109911;
        double r5109913 = sqrt(r5109912);
        double r5109914 = r5109898 * r5109913;
        return r5109914;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5109915 = 1.0;
        double r5109916 = D;
        double r5109917 = M;
        double r5109918 = r5109916 * r5109917;
        double r5109919 = 2.0;
        double r5109920 = r5109918 / r5109919;
        double r5109921 = d;
        double r5109922 = r5109920 / r5109921;
        double r5109923 = h;
        double r5109924 = cbrt(r5109923);
        double r5109925 = r5109924 * r5109924;
        double r5109926 = r5109915 / r5109925;
        double r5109927 = r5109922 / r5109926;
        double r5109928 = l;
        double r5109929 = r5109922 / r5109928;
        double r5109930 = r5109915 / r5109924;
        double r5109931 = r5109929 / r5109930;
        double r5109932 = r5109927 * r5109931;
        double r5109933 = r5109915 - r5109932;
        double r5109934 = sqrt(r5109933);
        double r5109935 = w0;
        double r5109936 = r5109934 * r5109935;
        return r5109936;
}

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 13.7

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

  2. Simplified13.2

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

  4. Applied add-cube-cbrt13.2

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

  5. Applied *-un-lft-identity13.2

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

  6. Applied times-frac13.2

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

  7. Applied times-frac8.9

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

  9. Applied div-inv8.9

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

  10. Applied associate-/r*8.2

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

  11. Final simplification8.2

    \[\leadsto \sqrt{1 - \frac{\frac{\frac{D \cdot M}{2}}{d}}{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \frac{\frac{\frac{\frac{D \cdot M}{2}}{d}}{\ell}}{\frac{1}{\sqrt[3]{h}}}} \cdot w0\]

Reproduce

herbie shell --seed 2019163 +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))))))