Average Error: 26.5 → 14.0
Time: 35.8s
Precision: binary64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -2.3326021187924278 \cdot 10^{-75}:\\ \;\;\;\;\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \le 4.334478490926267 \cdot 10^{-114}:\\ \;\;\;\;\left(\sqrt[3]{\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)} \cdot \sqrt[3]{\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)}\right) \cdot \sqrt[3]{\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]
\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
↓
\begin{array}{l}
\mathbf{if}\;\ell \le -2.3326021187924278 \cdot 10^{-75}:\\
\;\;\;\;\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\

\mathbf{elif}\;\ell \le 4.334478490926267 \cdot 10^{-114}:\\
\;\;\;\;\left(\sqrt[3]{\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)} \cdot \sqrt[3]{\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)}\right) \cdot \sqrt[3]{\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\

\end{array}
double code(double d, double h, double l, double M, double D) {
	return ((double) (((double) (((double) pow(((double) (d / h)), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (d / l)), ((double) (1.0 / 2.0)))))) * ((double) (1.0 - ((double) (((double) (((double) (1.0 / 2.0)) * ((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)))) * ((double) (h / l))))))));
}
↓
double code(double d, double h, double l, double M, double D) {
	double VAR;
	if ((l <= -2.332602118792428e-75)) {
		VAR = ((double) (((double) (((double) (((double) pow(((double) (1.0 / ((double) (((double) cbrt(h)) * ((double) cbrt(h)))))), ((double) (1.0 / 2.0)))) * ((double) (((double) pow(((double) (1.0 / ((double) cbrt(((double) (((double) cbrt(h)) * ((double) cbrt(h)))))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (d / ((double) cbrt(((double) cbrt(h)))))), ((double) (1.0 / 2.0)))))))) * ((double) (((double) pow(((double) (((double) (((double) cbrt(d)) * ((double) cbrt(d)))) / ((double) (((double) cbrt(l)) * ((double) cbrt(l)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(d)) / ((double) cbrt(l)))), ((double) (1.0 / 2.0)))))))) * ((double) (1.0 - ((double) (((double) (((double) (1.0 / 2.0)) * ((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)))) * ((double) (h / l))))))));
	} else {
		double VAR_1;
		if ((l <= 4.334478490926267e-114)) {
			VAR_1 = ((double) (((double) (((double) cbrt(((double) (((double) (((double) (((double) pow(((double) (1.0 / ((double) (((double) cbrt(h)) * ((double) cbrt(h)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (d / ((double) cbrt(h)))), ((double) (1.0 / 2.0)))))) * ((double) (((double) pow(((double) (((double) (((double) cbrt(d)) * ((double) cbrt(d)))) / ((double) (((double) cbrt(l)) * ((double) cbrt(l)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(d)) / ((double) cbrt(l)))), ((double) (1.0 / 2.0)))))))) * ((double) (1.0 - ((double) (((double) (((double) (1.0 * ((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)))) * h)) / ((double) (2.0 * l)))))))))) * ((double) cbrt(((double) (((double) (((double) (((double) pow(((double) (1.0 / ((double) (((double) cbrt(h)) * ((double) cbrt(h)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (d / ((double) cbrt(h)))), ((double) (1.0 / 2.0)))))) * ((double) (((double) pow(((double) (((double) (((double) cbrt(d)) * ((double) cbrt(d)))) / ((double) (((double) cbrt(l)) * ((double) cbrt(l)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(d)) / ((double) cbrt(l)))), ((double) (1.0 / 2.0)))))))) * ((double) (1.0 - ((double) (((double) (((double) (1.0 * ((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)))) * h)) / ((double) (2.0 * l)))))))))))) * ((double) cbrt(((double) (((double) (((double) (((double) pow(((double) (1.0 / ((double) (((double) cbrt(h)) * ((double) cbrt(h)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (d / ((double) cbrt(h)))), ((double) (1.0 / 2.0)))))) * ((double) (((double) pow(((double) (((double) (((double) cbrt(d)) * ((double) cbrt(d)))) / ((double) (((double) cbrt(l)) * ((double) cbrt(l)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(d)) / ((double) cbrt(l)))), ((double) (1.0 / 2.0)))))))) * ((double) (1.0 - ((double) (((double) (((double) (1.0 * ((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)))) * h)) / ((double) (2.0 * l))))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) pow(d, ((double) (1.0 / 2.0)))) * ((double) pow(((double) (1.0 / h)), ((double) (1.0 / 2.0)))))) * ((double) (((double) pow(((double) (((double) (((double) cbrt(d)) * ((double) cbrt(d)))) / ((double) (((double) cbrt(l)) * ((double) cbrt(l)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(d)) / ((double) cbrt(l)))), ((double) (1.0 / 2.0)))))))) * ((double) (1.0 - ((double) (((double) (((double) (1.0 / 2.0)) * ((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)))) * ((double) (h / l))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if l < -2.3326021187924278e-75

    1. Initial program 25.0

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt25.2

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

    4. Applied add-cube-cbrt25.3

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

    5. Applied times-frac25.3

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

    6. Applied unpow-prod-down21.9

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

    8. Applied add-cube-cbrt22.0

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

    9. Applied *-un-lft-identity22.0

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

    10. Applied times-frac22.0

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

    11. Applied unpow-prod-down16.2

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

    13. Applied add-cube-cbrt16.2

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

    14. Applied cbrt-prod16.2

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

    15. Applied *-un-lft-identity16.2

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

    16. Applied times-frac16.2

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

    17. Applied unpow-prod-down15.2

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

    if -2.3326021187924278e-75 < l < 4.334478490926267e-114

    1. Initial program 31.4

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt31.7

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

    4. Applied add-cube-cbrt31.8

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

    5. Applied times-frac31.7

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

    6. Applied unpow-prod-down25.2

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

    8. Applied add-cube-cbrt25.2

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

    9. Applied *-un-lft-identity25.2

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

    10. Applied times-frac25.2

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

    11. Applied unpow-prod-down24.0

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

    13. Applied associate-*l/24.0

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

    14. Applied frac-times10.9

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

    16. Applied add-cube-cbrt11.2

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

    if 4.334478490926267e-114 < l

    1. Initial program 25.4

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt25.6

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

    4. Applied add-cube-cbrt25.8

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

    5. Applied times-frac25.8

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

    6. Applied unpow-prod-down22.1

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

    8. Applied div-inv22.1

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

    9. Applied unpow-prod-down14.5

      \[\leadsto \left(\color{blue}{\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.3326021187924278 \cdot 10^{-75}:\\ \;\;\;\;\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;\ell \le 4.334478490926267 \cdot 10^{-114}:\\ \;\;\;\;\left(\sqrt[3]{\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)} \cdot \sqrt[3]{\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)}\right) \cdot \sqrt[3]{\left(\left({\left(\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\left(1 \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({d}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))