Henrywood and Agarwal, Equation (12)

Average Error: 26.5 → 14.7

Time: 22.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;h \le -5.25026829655488903 \cdot 10^{-253}:\\ \;\;\;\;\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{h}{\sqrt[3]{\ell}}\right)\\ \mathbf{elif}\;h \le 4.8325147382386977 \cdot 10^{-211}:\\ \;\;\;\;e^{\frac{1}{2} \cdot \left(\log \left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right) \cdot 3 + 3 \cdot \log \left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)\right) + \log \left(1 - \frac{1}{2} \cdot \left(\frac{h}{\sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{h}{\sqrt[3]{\ell}} \cdot \left(\frac{1}{2} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\sqrt[3]{\ell}}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}}\right)\right)\\ \end{array}\]

C

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 ((h <= -5.250268296554889e-253)) {
		VAR = ((double) (((double) (((double) (((double) pow(((double) (((double) (((double) cbrt(d)) / ((double) cbrt(h)))) * ((double) (((double) cbrt(d)) / ((double) cbrt(h)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(d)) / ((double) cbrt(h)))), ((double) (1.0 / 2.0)))))) * ((double) (((double) pow(((double) (((double) (((double) cbrt(d)) / ((double) cbrt(l)))) * ((double) (((double) cbrt(d)) / ((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) (((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 2.0)))) / ((double) cbrt(l)))) * ((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 2.0)))) / ((double) cbrt(l)))))))) * ((double) (h / ((double) cbrt(l))))))))));
	} else {
		double VAR_1;
		if ((h <= 4.832514738238698e-211)) {
			VAR_1 = ((double) exp(((double) (((double) (((double) (1.0 / 2.0)) * ((double) (((double) (((double) log(((double) (((double) cbrt(d)) / ((double) cbrt(h)))))) * 3.0)) + ((double) (3.0 * ((double) log(((double) (((double) cbrt(d)) / ((double) cbrt(l)))))))))))) + ((double) log(((double) (1.0 - ((double) (((double) (1.0 / 2.0)) * ((double) (((double) (h / ((double) cbrt(l)))) * ((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), 2.0)) / ((double) (((double) cbrt(l)) * ((double) cbrt(l))))))))))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) pow(((double) (((double) (((double) cbrt(d)) / ((double) cbrt(h)))) * ((double) (((double) cbrt(d)) / ((double) cbrt(h)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(d)) / ((double) cbrt(h)))), ((double) (1.0 / 2.0)))))) * ((double) (((double) pow(((double) (((double) (((double) cbrt(d)) / ((double) cbrt(l)))) * ((double) (((double) cbrt(d)) / ((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) (h / ((double) cbrt(l)))) * ((double) (((double) (1.0 / 2.0)) * ((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 2.0)))) / ((double) (((double) cbrt(l)) / ((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 2.0)))) / ((double) cbrt(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

Derivation

1. Split input into 3 regimes

2. if h < -5.25026829655488903e-253

   1. Initial program 26.2

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

   3. Applied add-cube-cbrt 26.4

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

   4. Applied add-cube-cbrt 26.5

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

   5. Applied times-frac 26.5

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

   6. Applied unpow-prod-down 21.8

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

   7. Simplified 21.8

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

   9. Applied add-cube-cbrt 21.8

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

   10. Applied *-un-lft-identity 21.8

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

   11. Applied times-frac 21.8

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

   12. Applied associate-*r* 20.4

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

   13. Simplified 20.6

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

   15. Applied add-cube-cbrt 20.7

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

   16. Applied add-cube-cbrt 20.8

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

   17. Applied times-frac 20.8

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

   18. Applied unpow-prod-down 15.9

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

   19. Simplified 15.9

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

   21. Applied sqr-pow 15.9

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

   22. Applied times-frac 14.5

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

   if -5.25026829655488903e-253 < h < 4.8325147382386977e-211

   1. Initial program 37.6

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

   3. Applied add-cube-cbrt 37.8

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

   4. Applied add-cube-cbrt 37.9

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

   5. Applied times-frac 37.9

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

   6. Applied unpow-prod-down 23.4

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

   7. Simplified 23.4

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

   9. Applied add-cube-cbrt 23.5

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

   10. Applied *-un-lft-identity 23.5

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

   11. Applied times-frac 23.5

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

   12. Applied associate-*r* 24.8

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

   13. Simplified 25.3

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

   15. Applied add-cube-cbrt 25.4

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

   16. Applied add-cube-cbrt 25.5

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

   17. Applied times-frac 25.5

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

   18. Applied unpow-prod-down 21.8

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

   19. Simplified 21.8

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

   21. Applied add-exp-log 23.0

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

   22. Applied add-exp-log 36.0

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

   23. Applied add-exp-log 36.2

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

   24. Applied div-exp 36.3

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

   25. Applied pow-exp 36.3

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

   26. Applied add-exp-log 36.6

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

   27. Applied add-exp-log 36.7

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

   28. Applied div-exp 36.7

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

   29. Applied add-exp-log 36.9

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

   30. Applied add-exp-log 36.9

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

   31. Applied div-exp 37.0

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

   32. Applied prod-exp 37.0

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

   33. Applied pow-exp 36.6

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

   34. Applied prod-exp 36.7

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

   35. Applied add-exp-log 36.7

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

   36. Applied add-exp-log 36.8

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

   37. Applied div-exp 36.8

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

   38. Applied pow-exp 36.8

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

   39. Applied add-exp-log 36.9

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

   40. Applied add-exp-log 36.9

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

   41. Applied div-exp 37.0

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

   42. Applied add-exp-log 37.0

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

   43. Applied add-exp-log 37.1

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

   44. Applied div-exp 37.1

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

   45. Applied prod-exp 37.1

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

   46. Applied pow-exp 35.1

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

   47. Applied prod-exp 35.2

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

   48. Applied prod-exp 35.3

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

   49. Applied prod-exp 35.3

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

   50. Simplified 21.8

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

   if 4.8325147382386977e-211 < h

   1. Initial program 23.8

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

   3. Applied add-cube-cbrt 24.0

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

   4. Applied add-cube-cbrt 24.1

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

   5. Applied times-frac 24.1

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

   6. Applied unpow-prod-down 20.3

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

   7. Simplified 20.3

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

   9. Applied add-cube-cbrt 20.3

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

   10. Applied *-un-lft-identity 20.3

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

   11. Applied times-frac 20.3

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

   12. Applied associate-*r* 18.9

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

   13. Simplified 18.7

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

   15. Applied add-cube-cbrt 18.8

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

   16. Applied add-cube-cbrt 19.0

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

   17. Applied times-frac 19.0

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

   18. Applied unpow-prod-down 14.3

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

   19. Simplified 14.3

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

   21. Applied sqr-pow 14.3

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

   22. Applied associate-/l* 12.8

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

   23. Simplified 12.8

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

4. Final simplification 14.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -5.25026829655488903 \cdot 10^{-253}:\\ \;\;\;\;\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{h}{\sqrt[3]{\ell}}\right)\\ \mathbf{elif}\;h \le 4.8325147382386977 \cdot 10^{-211}:\\ \;\;\;\;e^{\frac{1}{2} \cdot \left(\log \left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right) \cdot 3 + 3 \cdot \log \left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)\right) + \log \left(1 - \frac{1}{2} \cdot \left(\frac{h}{\sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{h}{\sqrt[3]{\ell}} \cdot \left(\frac{1}{2} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\sqrt[3]{\ell}}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))))