Average Error: 26.3 → 15.0
Time: 18.4s
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 \leq -1.0934712194816513 \cdot 10^{+30}:\\ \;\;\;\;\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\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) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -9.103863212509198 \cdot 10^{-54}:\\ \;\;\;\;0.125 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5}\right) \cdot {\left(\frac{1}{{-1}^{4} \cdot {d}^{1}}\right)}^{1}\right) - 1 \cdot \left(d \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\ \mathbf{elif}\;\ell \leq 2.3451948097595518 \cdot 10^{+33}:\\ \;\;\;\;\left(\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right)\right) \cdot \left(1 - \frac{h \cdot \left(1 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell \cdot 2}\right)\\ \mathbf{elif}\;\ell \leq 6.009050044890221 \cdot 10^{+124}:\\ \;\;\;\;1 \cdot \left(d \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right) - 0.125 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5}\right) \cdot {\left(\frac{1}{{d}^{1}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\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 + \frac{0}{\ell \cdot 2}\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 \leq -1.0934712194816513 \cdot 10^{+30}:\\
\;\;\;\;\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\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) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\\

\mathbf{elif}\;\ell \leq -9.103863212509198 \cdot 10^{-54}:\\
\;\;\;\;0.125 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5}\right) \cdot {\left(\frac{1}{{-1}^{4} \cdot {d}^{1}}\right)}^{1}\right) - 1 \cdot \left(d \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\

\mathbf{elif}\;\ell \leq 2.3451948097595518 \cdot 10^{+33}:\\
\;\;\;\;\left(\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right)\right) \cdot \left(1 - \frac{h \cdot \left(1 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell \cdot 2}\right)\\

\mathbf{elif}\;\ell \leq 6.009050044890221 \cdot 10^{+124}:\\
\;\;\;\;1 \cdot \left(d \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right) - 0.125 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5}\right) \cdot {\left(\frac{1}{{d}^{1}}\right)}^{1}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\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 + \frac{0}{\ell \cdot 2}\right)\\

\end{array}
(FPCore (d h l M D)
 :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)))))
↓
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.0934712194816513e+30)
   (*
    (*
     (pow (* (cbrt d) (cbrt d)) (/ 1.0 2.0))
     (pow (/ (cbrt d) h) (/ 1.0 2.0)))
    (*
     (*
      (pow (/ 1.0 (* (cbrt l) (cbrt l))) (/ 1.0 2.0))
      (pow (/ d (cbrt l)) (/ 1.0 2.0)))
     (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l)))))
   (if (<= l -9.103863212509198e-54)
     (-
      (*
       0.125
       (*
        (* (* (* M D) (* M D)) (pow (/ (pow h 1.0) (pow l 3.0)) 0.5))
        (pow (/ 1.0 (* (pow -1.0 4.0) (pow d 1.0))) 1.0)))
      (* 1.0 (* d (pow (/ 1.0 (* (pow h 1.0) (pow l 1.0))) 0.5))))
     (if (<= l 2.3451948097595518e+33)
       (*
        (*
         (*
          (pow (* (cbrt d) (cbrt d)) (/ 1.0 2.0))
          (pow (/ (cbrt d) h) (/ 1.0 2.0)))
         (*
          (pow (/ 1.0 (* (cbrt l) (cbrt l))) (/ 1.0 2.0))
          (*
           (pow
            (/ (* (cbrt d) (cbrt d)) (cbrt (* (cbrt l) (cbrt l))))
            (/ 1.0 2.0))
           (pow (/ (cbrt d) (cbrt (cbrt l))) (/ 1.0 2.0)))))
        (- 1.0 (/ (* h (* 1.0 (pow (/ (* M D) (* d 2.0)) 2.0))) (* l 2.0))))
       (if (<= l 6.009050044890221e+124)
         (-
          (* 1.0 (* d (pow (/ 1.0 (* (pow h 1.0) (pow l 1.0))) 0.5)))
          (*
           0.125
           (*
            (* (* (* M D) (* M D)) (pow (/ (pow h 1.0) (pow l 3.0)) 0.5))
            (pow (/ 1.0 (pow d 1.0)) 1.0))))
         (*
          (*
           (*
            (pow (* (cbrt d) (cbrt d)) (/ 1.0 2.0))
            (pow (/ (cbrt d) h) (/ 1.0 2.0)))
           (*
            (pow (/ 1.0 (* (cbrt l) (cbrt l))) (/ 1.0 2.0))
            (pow (/ d (cbrt l)) (/ 1.0 2.0))))
          (+ 1.0 (/ 0.0 (* l 2.0)))))))))
double code(double d, double h, double l, double M, double D) {
	return ((double) (((double) (((double) pow((d / h), (1.0 / 2.0))) * ((double) pow((d / l), (1.0 / 2.0))))) * ((double) (1.0 - ((double) (((double) ((1.0 / 2.0) * ((double) pow((((double) (M * D)) / ((double) (2.0 * d))), 2.0)))) * (h / l)))))));
}
↓
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if ((l <= -1.0934712194816513e+30)) {
		tmp = ((double) (((double) (((double) pow(((double) (((double) cbrt(d)) * ((double) cbrt(d)))), (1.0 / 2.0))) * ((double) pow((((double) cbrt(d)) / h), (1.0 / 2.0))))) * ((double) (((double) (((double) pow((1.0 / ((double) (((double) cbrt(l)) * ((double) cbrt(l))))), (1.0 / 2.0))) * ((double) pow((d / ((double) cbrt(l))), (1.0 / 2.0))))) * ((double) (1.0 - ((double) (((double) ((1.0 / 2.0) * ((double) pow((((double) (M * D)) / ((double) (d * 2.0))), 2.0)))) * (h / l)))))))));
	} else {
		double tmp_1;
		if ((l <= -9.103863212509198e-54)) {
			tmp_1 = ((double) (((double) (0.125 * ((double) (((double) (((double) (((double) (M * D)) * ((double) (M * D)))) * ((double) pow((((double) pow(h, 1.0)) / ((double) pow(l, 3.0))), 0.5)))) * ((double) pow((1.0 / ((double) (((double) pow(-1.0, 4.0)) * ((double) pow(d, 1.0))))), 1.0)))))) - ((double) (1.0 * ((double) (d * ((double) pow((1.0 / ((double) (((double) pow(h, 1.0)) * ((double) pow(l, 1.0))))), 0.5))))))));
		} else {
			double tmp_2;
			if ((l <= 2.3451948097595518e+33)) {
				tmp_2 = ((double) (((double) (((double) (((double) pow(((double) (((double) cbrt(d)) * ((double) cbrt(d)))), (1.0 / 2.0))) * ((double) pow((((double) cbrt(d)) / h), (1.0 / 2.0))))) * ((double) (((double) pow((1.0 / ((double) (((double) cbrt(l)) * ((double) cbrt(l))))), (1.0 / 2.0))) * ((double) (((double) pow((((double) (((double) cbrt(d)) * ((double) cbrt(d)))) / ((double) cbrt(((double) (((double) cbrt(l)) * ((double) cbrt(l))))))), (1.0 / 2.0))) * ((double) pow((((double) cbrt(d)) / ((double) cbrt(((double) cbrt(l))))), (1.0 / 2.0))))))))) * ((double) (1.0 - (((double) (h * ((double) (1.0 * ((double) pow((((double) (M * D)) / ((double) (d * 2.0))), 2.0)))))) / ((double) (l * 2.0)))))));
			} else {
				double tmp_3;
				if ((l <= 6.009050044890221e+124)) {
					tmp_3 = ((double) (((double) (1.0 * ((double) (d * ((double) pow((1.0 / ((double) (((double) pow(h, 1.0)) * ((double) pow(l, 1.0))))), 0.5)))))) - ((double) (0.125 * ((double) (((double) (((double) (((double) (M * D)) * ((double) (M * D)))) * ((double) pow((((double) pow(h, 1.0)) / ((double) pow(l, 3.0))), 0.5)))) * ((double) pow((1.0 / ((double) pow(d, 1.0))), 1.0))))))));
				} else {
					tmp_3 = ((double) (((double) (((double) (((double) pow(((double) (((double) cbrt(d)) * ((double) cbrt(d)))), (1.0 / 2.0))) * ((double) pow((((double) cbrt(d)) / h), (1.0 / 2.0))))) * ((double) (((double) pow((1.0 / ((double) (((double) cbrt(l)) * ((double) cbrt(l))))), (1.0 / 2.0))) * ((double) pow((d / ((double) cbrt(l))), (1.0 / 2.0))))))) * ((double) (1.0 + (0.0 / ((double) (l * 2.0)))))));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

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 5 regimes

  2. if l < -1.09347121948165134e30

    1. Initial program 25.3

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

    3. Applied *-un-lft-identity_binary6425.3

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

    4. Applied add-cube-cbrt_binary6425.6

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

    5. Applied times-frac_binary6425.6

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

    6. Applied unpow-prod-down_binary6420.7

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

    7. Simplified20.7

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

    9. Applied add-cube-cbrt_binary6420.8

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

    10. Applied *-un-lft-identity_binary6420.8

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

    11. Applied times-frac_binary6420.8

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

    12. Applied unpow-prod-down_binary6417.1

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

    14. Applied associate-*l*_binary6416.5

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

    if -1.09347121948165134e30 < l < -9.10386321250919751e-54

    1. Initial program 19.0

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

    3. Applied *-un-lft-identity_binary6419.0

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

    4. Applied add-cube-cbrt_binary6419.4

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

    5. Applied times-frac_binary6419.4

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

    6. Applied unpow-prod-down_binary6415.1

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

    7. Simplified15.1

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

    9. Applied add-cube-cbrt_binary6415.3

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

    10. Applied *-un-lft-identity_binary6415.3

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

    11. Applied times-frac_binary6415.3

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

    12. Applied unpow-prod-down_binary6414.6

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

    13. Taylor expanded around -inf 27.0

      \[\leadsto \color{blue}{0.125 \cdot \left({\left(\frac{1}{{d}^{1} \cdot {-1}^{4}}\right)}^{1} \cdot \left({\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5} \cdot \left({D}^{2} \cdot {M}^{2}\right)\right)\right) - 1 \cdot \left(d \cdot {\left(\frac{1}{{\ell}^{1} \cdot {h}^{1}}\right)}^{0.5}\right)}\]

    14. Simplified11.0

      \[\leadsto \color{blue}{0.125 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5}\right) \cdot {\left(\frac{1}{{-1}^{4} \cdot {d}^{1}}\right)}^{1}\right) - 1 \cdot \left(d \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)}\]

    if -9.10386321250919751e-54 < l < 2.34519480975955178e33

    1. Initial program 27.3

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

    3. Applied *-un-lft-identity_binary6427.3

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

    4. Applied add-cube-cbrt_binary6427.5

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

    5. Applied times-frac_binary6427.5

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

    6. Applied unpow-prod-down_binary6425.2

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

    7. Simplified25.2

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

    9. Applied add-cube-cbrt_binary6425.3

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

    10. Applied *-un-lft-identity_binary6425.3

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

    11. Applied times-frac_binary6425.3

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

    12. Applied unpow-prod-down_binary6421.3

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

    14. Applied associate-*l/_binary6421.3

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

    15. Applied frac-times_binary6412.0

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

    16. Simplified12.0

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

    18. Applied add-cube-cbrt_binary6412.0

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

    19. Applied cbrt-prod_binary6412.1

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

    20. Applied add-cube-cbrt_binary6412.3

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

    21. Applied times-frac_binary6412.3

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

    22. Applied unpow-prod-down_binary6410.2

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

    if 2.34519480975955178e33 < l < 6.0090500448902209e124

    1. Initial program 23.0

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

    2. Taylor expanded around inf 29.2

      \[\leadsto \color{blue}{1 \cdot \left(d \cdot {\left(\frac{1}{{\ell}^{1} \cdot {h}^{1}}\right)}^{0.5}\right) - 0.125 \cdot \left({\left(\frac{1}{{d}^{1}}\right)}^{1} \cdot \left(\left({M}^{2} \cdot {D}^{2}\right) \cdot {\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5}\right)\right)}\]

    3. Simplified15.8

      \[\leadsto \color{blue}{1 \cdot \left(d \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right) - 0.125 \cdot \left({\left(\frac{1}{{d}^{1}}\right)}^{1} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5}\right)\right)}\]

    if 6.0090500448902209e124 < l

    1. Initial program 29.8

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

    3. Applied *-un-lft-identity_binary6429.8

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

    4. Applied add-cube-cbrt_binary6430.0

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

    5. Applied times-frac_binary6430.0

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

    6. Applied unpow-prod-down_binary6424.7

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

    7. Simplified24.7

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

    9. Applied add-cube-cbrt_binary6424.8

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

    10. Applied *-un-lft-identity_binary6424.8

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

    11. Applied times-frac_binary6424.8

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

    12. Applied unpow-prod-down_binary6419.8

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

    14. Applied associate-*l/_binary6419.8

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

    15. Applied frac-times_binary6421.8

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

    16. Simplified21.8

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

    17. Taylor expanded around 0 21.7

      \[\leadsto \left(\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \frac{\color{blue}{0}}{2 \cdot \ell}\right)\]
  3. Recombined 5 regimes into one program.

  4. Final simplification15.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.0934712194816513 \cdot 10^{+30}:\\ \;\;\;\;\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\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) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq -9.103863212509198 \cdot 10^{-54}:\\ \;\;\;\;0.125 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5}\right) \cdot {\left(\frac{1}{{-1}^{4} \cdot {d}^{1}}\right)}^{1}\right) - 1 \cdot \left(d \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right)\\ \mathbf{elif}\;\ell \leq 2.3451948097595518 \cdot 10^{+33}:\\ \;\;\;\;\left(\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)}^{\left(\frac{1}{2}\right)}\right)\right)\right) \cdot \left(1 - \frac{h \cdot \left(1 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell \cdot 2}\right)\\ \mathbf{elif}\;\ell \leq 6.009050044890221 \cdot 10^{+124}:\\ \;\;\;\;1 \cdot \left(d \cdot {\left(\frac{1}{{h}^{1} \cdot {\ell}^{1}}\right)}^{0.5}\right) - 0.125 \cdot \left(\left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot {\left(\frac{{h}^{1}}{{\ell}^{3}}\right)}^{0.5}\right) \cdot {\left(\frac{1}{{d}^{1}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{h}\right)}^{\left(\frac{1}{2}\right)}\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 + \frac{0}{\ell \cdot 2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))))