Average Error: 13.1 → 7.8
Time: 8.7m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\left|\sqrt{1 - \left(\frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot M}{\frac{2}{\frac{D}{d}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot M}{\frac{2}{\frac{D}{d}}}}\right| \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\left|\sqrt{1 - \left(\frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot M}{\frac{2}{\frac{D}{d}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot M}{\frac{2}{\frac{D}{d}}}}\right| \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r92036908 = w0;
        double r92036909 = 1.0;
        double r92036910 = M;
        double r92036911 = D;
        double r92036912 = r92036910 * r92036911;
        double r92036913 = 2.0;
        double r92036914 = d;
        double r92036915 = r92036913 * r92036914;
        double r92036916 = r92036912 / r92036915;
        double r92036917 = pow(r92036916, r92036913);
        double r92036918 = h;
        double r92036919 = l;
        double r92036920 = r92036918 / r92036919;
        double r92036921 = r92036917 * r92036920;
        double r92036922 = r92036909 - r92036921;
        double r92036923 = sqrt(r92036922);
        double r92036924 = r92036908 * r92036923;
        return r92036924;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r92036925 = 1.0;
        double r92036926 = h;
        double r92036927 = cbrt(r92036926);
        double r92036928 = l;
        double r92036929 = cbrt(r92036928);
        double r92036930 = r92036927 / r92036929;
        double r92036931 = M;
        double r92036932 = r92036930 * r92036931;
        double r92036933 = 2.0;
        double r92036934 = D;
        double r92036935 = d;
        double r92036936 = r92036934 / r92036935;
        double r92036937 = r92036933 / r92036936;
        double r92036938 = r92036932 / r92036937;
        double r92036939 = r92036938 * r92036930;
        double r92036940 = r92036939 * r92036938;
        double r92036941 = r92036925 - r92036940;
        double r92036942 = sqrt(r92036941);
        double r92036943 = fabs(r92036942);
        double r92036944 = w0;
        double r92036945 = r92036943 * r92036944;
        return r92036945;
}

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 13.1

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

  2. Simplified13.1

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

  4. Applied add-cube-cbrt13.2

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

  5. Applied add-cube-cbrt13.2

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

  6. Applied times-frac13.2

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

  7. Applied associate-*r*10.1

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

  9. Applied add-sqr-sqrt10.1

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

  10. Applied rem-sqrt-square10.1

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

  11. Simplified8.2

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

  13. Applied associate-*l*7.8

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

  14. Final simplification7.8

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

Reproduce

herbie shell --seed 2019124 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))