Average Error: 26.9 → 19.8
Time: 24.7s
Precision: binary64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -2.14251568308186794 \cdot 10^{51}:\\ \;\;\;\;\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{d}{\ell}} \cdot \sqrt[3]{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \frac{1}{\ell}\right)\\ \mathbf{elif}\;d \le 15621975732124.5645:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\left({\left(\sqrt[3]{d} \cdot \frac{\sqrt[3]{d}}{\sqrt{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]
\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}\;d \le -2.14251568308186794 \cdot 10^{51}:\\
\;\;\;\;\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(\sqrt[3]{\frac{d}{\ell}} \cdot \sqrt[3]{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \frac{1}{\ell}\right)\\

\mathbf{elif}\;d \le 15621975732124.5645:\\
\;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\left({\left(\sqrt[3]{d} \cdot \frac{\sqrt[3]{d}}{\sqrt{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\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 ((d <= -2.142515683081868e+51)) {
		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) cbrt(((double) (d / l)))) * ((double) cbrt(((double) (d / l)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(((double) (((double) (((double) cbrt(d)) / ((double) cbrt(l)))) * ((double) (((double) cbrt(d)) / ((double) cbrt(l)))))))) * ((double) cbrt(((double) (((double) cbrt(d)) / ((double) cbrt(l)))))))), ((double) (1.0 / 2.0)))))))) * ((double) (1.0 - ((double) (((double) (((double) (1.0 / 2.0)) * ((double) (h * ((double) pow(((double) (D * ((double) (M / ((double) (d * 2.0)))))), 2.0)))))) * ((double) (1.0 / l))))))));
	} else {
		double VAR_1;
		if ((d <= 15621975732124.564)) {
			VAR_1 = ((double) (((double) (((double) pow(((double) (d / h)), ((double) (1.0 / 2.0)))) * ((double) (((double) pow(((double) (1.0 / ((double) (((double) cbrt(l)) * ((double) cbrt(l)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (d / ((double) cbrt(l)))), ((double) (1.0 / 2.0)))))))) * ((double) (1.0 - ((double) (((double) (((double) (1.0 / 2.0)) * ((double) pow(((double) (((double) (D * M)) / ((double) (d * 2.0)))), 2.0)))) * ((double) (h / l))))))));
		} else {
			VAR_1 = ((double) (((double) (1.0 - ((double) (((double) (((double) (1.0 / 2.0)) * ((double) pow(((double) (((double) (D * M)) / ((double) (d * 2.0)))), 2.0)))) * ((double) (h / l)))))) * ((double) (((double) (((double) pow(((double) (((double) cbrt(d)) * ((double) (((double) cbrt(d)) / ((double) sqrt(h)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(d)) / ((double) sqrt(h)))), ((double) (1.0 / 2.0)))))) * ((double) pow(((double) (d / l)), ((double) (1.0 / 2.0))))))));
		}
		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 d < -2.14251568308186794e51

    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-cbrt26.5

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

    4. Applied add-cube-cbrt26.6

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

    5. Applied times-frac26.6

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

    6. Applied unpow-prod-down16.0

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

    7. Simplified16.0

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

    9. Applied add-cube-cbrt16.1

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

    10. Applied unpow-prod-down16.1

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

    12. Applied div-inv16.1

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

    13. Applied associate-*r*15.2

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

    14. Simplified14.6

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

    16. Applied add-cube-cbrt14.6

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

    17. Applied add-cube-cbrt14.7

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

    18. Applied times-frac14.7

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

    19. Applied cbrt-prod14.7

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

    20. Simplified14.7

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

    if -2.14251568308186794e51 < d < 15621975732124.5645

    1. Initial program 28.8

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

    3. Applied add-cube-cbrt29.1

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

    4. Applied *-un-lft-identity29.1

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

    5. Applied times-frac29.1

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

    6. Applied unpow-prod-down25.0

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

    if 15621975732124.5645 < d

    1. Initial program 23.3

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

    3. Applied add-sqr-sqrt23.3

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

    4. Applied add-cube-cbrt23.6

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

    5. Applied times-frac23.6

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

    6. Applied unpow-prod-down13.0

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

    7. Simplified13.0

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

  4. Final simplification19.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -2.14251568308186794 \cdot 10^{51}:\\ \;\;\;\;\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{\frac{d}{\ell}} \cdot \sqrt[3]{\frac{d}{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt[3]{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \left(h \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \frac{1}{\ell}\right)\\ \mathbf{elif}\;d \le 15621975732124.5645:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\left({\left(\sqrt[3]{d} \cdot \frac{\sqrt[3]{d}}{\sqrt{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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)))))