Average Error: 13.4 → 8.1
Time: 24.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\left(\sqrt{\sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{D \cdot M}{2 \cdot d}}} \cdot w0\right) \cdot \sqrt{\sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{D \cdot M}{2 \cdot d}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\left(\sqrt{\sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{D \cdot M}{2 \cdot d}}} \cdot w0\right) \cdot \sqrt{\sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{D \cdot M}{2 \cdot d}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r2557838 = w0;
        double r2557839 = 1.0;
        double r2557840 = M;
        double r2557841 = D;
        double r2557842 = r2557840 * r2557841;
        double r2557843 = 2.0;
        double r2557844 = d;
        double r2557845 = r2557843 * r2557844;
        double r2557846 = r2557842 / r2557845;
        double r2557847 = pow(r2557846, r2557843);
        double r2557848 = h;
        double r2557849 = l;
        double r2557850 = r2557848 / r2557849;
        double r2557851 = r2557847 * r2557850;
        double r2557852 = r2557839 - r2557851;
        double r2557853 = sqrt(r2557852);
        double r2557854 = r2557838 * r2557853;
        return r2557854;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r2557855 = 1.0;
        double r2557856 = h;
        double r2557857 = cbrt(r2557856);
        double r2557858 = l;
        double r2557859 = cbrt(r2557858);
        double r2557860 = r2557857 / r2557859;
        double r2557861 = D;
        double r2557862 = M;
        double r2557863 = r2557861 * r2557862;
        double r2557864 = 2.0;
        double r2557865 = d;
        double r2557866 = r2557864 * r2557865;
        double r2557867 = r2557863 / r2557866;
        double r2557868 = r2557860 * r2557867;
        double r2557869 = r2557868 * r2557860;
        double r2557870 = r2557860 * r2557869;
        double r2557871 = r2557870 * r2557867;
        double r2557872 = r2557855 - r2557871;
        double r2557873 = sqrt(r2557872);
        double r2557874 = sqrt(r2557873);
        double r2557875 = w0;
        double r2557876 = r2557874 * r2557875;
        double r2557877 = r2557876 * r2557874;
        return r2557877;
}

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 13.4

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

  2. Simplified13.4

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

  4. Applied associate-*l*12.1

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

  6. Applied add-cube-cbrt12.1

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

  7. Applied add-cube-cbrt12.2

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

  8. Applied times-frac12.2

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

  9. Applied associate-*r*8.8

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

  10. Simplified8.1

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

  12. Applied add-sqr-sqrt8.1

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

  13. Applied sqrt-prod8.1

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

  14. Applied associate-*l*8.1

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

  15. Final simplification8.1

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

Reproduce

herbie shell --seed 2019128 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))