Average Error: 13.4 → 8.0
Time: 22.2s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\left(\sqrt{\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\sqrt[3]{\frac{D \cdot M}{2}} \cdot \sqrt[3]{\frac{D \cdot M}{2}}\right) \cdot \frac{\sqrt[3]{\frac{D \cdot M}{2}}}{d}\right)\right)\right)}} \cdot \sqrt{\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\sqrt[3]{\frac{D \cdot M}{2}} \cdot \sqrt[3]{\frac{D \cdot M}{2}}\right) \cdot \frac{\sqrt[3]{\frac{D \cdot M}{2}}}{d}\right)\right)\right)}}\right) \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\left(\sqrt{\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\sqrt[3]{\frac{D \cdot M}{2}} \cdot \sqrt[3]{\frac{D \cdot M}{2}}\right) \cdot \frac{\sqrt[3]{\frac{D \cdot M}{2}}}{d}\right)\right)\right)}} \cdot \sqrt{\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\sqrt[3]{\frac{D \cdot M}{2}} \cdot \sqrt[3]{\frac{D \cdot M}{2}}\right) \cdot \frac{\sqrt[3]{\frac{D \cdot M}{2}}}{d}\right)\right)\right)}}\right) \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r4162707 = w0;
        double r4162708 = 1.0;
        double r4162709 = M;
        double r4162710 = D;
        double r4162711 = r4162709 * r4162710;
        double r4162712 = 2.0;
        double r4162713 = d;
        double r4162714 = r4162712 * r4162713;
        double r4162715 = r4162711 / r4162714;
        double r4162716 = pow(r4162715, r4162712);
        double r4162717 = h;
        double r4162718 = l;
        double r4162719 = r4162717 / r4162718;
        double r4162720 = r4162716 * r4162719;
        double r4162721 = r4162708 - r4162720;
        double r4162722 = sqrt(r4162721);
        double r4162723 = r4162707 * r4162722;
        return r4162723;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r4162724 = 1.0;
        double r4162725 = h;
        double r4162726 = cbrt(r4162725);
        double r4162727 = l;
        double r4162728 = cbrt(r4162727);
        double r4162729 = r4162726 / r4162728;
        double r4162730 = D;
        double r4162731 = M;
        double r4162732 = r4162730 * r4162731;
        double r4162733 = 2.0;
        double r4162734 = r4162732 / r4162733;
        double r4162735 = d;
        double r4162736 = r4162734 / r4162735;
        double r4162737 = r4162736 * r4162729;
        double r4162738 = cbrt(r4162734);
        double r4162739 = r4162738 * r4162738;
        double r4162740 = r4162738 / r4162735;
        double r4162741 = r4162739 * r4162740;
        double r4162742 = r4162729 * r4162741;
        double r4162743 = r4162737 * r4162742;
        double r4162744 = r4162729 * r4162743;
        double r4162745 = r4162724 - r4162744;
        double r4162746 = sqrt(r4162745);
        double r4162747 = sqrt(r4162746);
        double r4162748 = r4162747 * r4162747;
        double r4162749 = w0;
        double r4162750 = r4162748 * r4162749;
        return r4162750;
}

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 add-cube-cbrt13.5

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

  5. Applied add-cube-cbrt13.5

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

  6. Applied times-frac13.5

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

  7. Applied associate-*r*10.3

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

  8. Simplified7.9

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

  10. Applied *-un-lft-identity7.9

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

  11. Applied add-cube-cbrt8.0

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

  12. Applied times-frac8.0

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

  14. Applied add-sqr-sqrt8.0

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

  15. Applied sqrt-prod8.0

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

  16. Final simplification8.0

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

Reproduce

herbie shell --seed 2019155 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))