Average Error: 26.9 → 14.6
Time: 20.3s
Precision: binary64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \leq 4.34839696867144 \cdot 10^{-28}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}{\sqrt{h} \cdot \left(\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)}\\ \end{array}\]
\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
↓
\begin{array}{l}
\mathbf{if}\;\ell \leq 4.34839696867144 \cdot 10^{-28}:\\
\;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\ell}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{d \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}{\sqrt{h} \cdot \left(\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\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 4.34839696867144e-28)
   (*
    (*
     (* (fabs (/ (cbrt d) (cbrt h))) (sqrt (/ (cbrt d) (cbrt h))))
     (* (sqrt (/ 1.0 (* (cbrt l) (cbrt l)))) (sqrt (/ d (cbrt l)))))
    (-
     1.0
     (*
      (*
       (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (* (cbrt h) (cbrt h)))
       (* (cbrt (cbrt h)) (cbrt (cbrt h))))
      (/ (cbrt (cbrt h)) l))))
   (/
    (* d (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l))))
    (* (sqrt h) (* (sqrt (* (cbrt l) (cbrt l))) (sqrt (cbrt l)))))))
double code(double d, double h, double l, double M, double D) {
	return (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)));
}
↓
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 4.34839696867144e-28) {
		tmp = ((fabs(cbrt(d) / cbrt(h)) * sqrt(cbrt(d) / cbrt(h))) * (sqrt(1.0 / (cbrt(l) * cbrt(l))) * sqrt(d / cbrt(l)))) * (1.0 - ((((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (cbrt(h) * cbrt(h))) * (cbrt(cbrt(h)) * cbrt(cbrt(h)))) * (cbrt(cbrt(h)) / l)));
	} else {
		tmp = (d * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)))) / (sqrt(h) * (sqrt(cbrt(l) * cbrt(l)) * sqrt(cbrt(l))));
	}
	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 2 regimes

  2. if l < 4.34839696867144036e-28

    1. Initial program 27.4

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

    2. Simplified27.4

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

    4. Applied add-cube-cbrt_binary6427.7

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

    5. Applied *-un-lft-identity_binary6427.7

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

    6. Applied times-frac_binary6427.7

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

    7. Applied sqrt-prod_binary6423.7

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    8. Using strategy rm

    9. Applied add-cube-cbrt_binary6423.8

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

    10. Applied add-cube-cbrt_binary6423.9

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

    11. Applied times-frac_binary6423.9

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

    12. Applied sqrt-prod_binary6419.2

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

    13. Simplified18.7

      \[\leadsto \left(\left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    14. Using strategy rm

    15. Applied *-un-lft-identity_binary6418.7

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

    16. Applied add-cube-cbrt_binary6418.7

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

    17. Applied times-frac_binary6418.7

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

    18. Applied associate-*r*_binary6414.8

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

    19. Simplified14.8

      \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\ell}\right)\]
    20. Using strategy rm

    21. Applied *-un-lft-identity_binary6414.8

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

    22. Applied add-cube-cbrt_binary6414.9

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

    23. Applied times-frac_binary6414.9

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

    24. Applied associate-*r*_binary6414.3

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

    25. Simplified14.3

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

    if 4.34839696867144036e-28 < l

    1. Initial program 25.9

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

    2. Simplified25.9

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

    4. Applied add-cube-cbrt_binary6426.1

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

    5. Applied *-un-lft-identity_binary6426.1

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

    6. Applied times-frac_binary6426.1

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

    7. Applied sqrt-prod_binary6422.8

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
    8. Using strategy rm

    9. Applied sqrt-div_binary6422.4

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

    10. Applied sqrt-div_binary6422.4

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

    11. Applied frac-times_binary6422.4

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

    12. Applied sqrt-div_binary6415.6

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

    13. Applied frac-times_binary6415.6

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

    14. Applied associate-*l/_binary6415.2

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

    15. Simplified15.1

      \[\leadsto \frac{\color{blue}{d \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}}{\sqrt{h} \cdot \left(\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.34839696867144 \cdot 10^{-28}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(\left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}{\sqrt{h} \cdot \left(\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020263 
(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)))))