Average Error: 26.2 → 15.4
Time: 28.0s
Precision: binary64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
↓
\[\begin{array}{l} t_0 := 0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\\ \mathbf{if}\;d \leq 2.1521460601640103 \cdot 10^{-293}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\\ \left(\left(\left|t_1\right| \cdot \sqrt{t_1}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{d}}}{\ell}}\right)\right)\right) \cdot \left(1 - t_0 \cdot \frac{h}{\ell}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{h \cdot t_0}{\ell}\right) \cdot \frac{d}{\left|\sqrt[3]{h}\right|}}{\sqrt{\sqrt[3]{h}} \cdot \sqrt{\ell}}\\ \end{array} \]
\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}
t_0 := 0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\\
\mathbf{if}\;d \leq 2.1521460601640103 \cdot 10^{-293}:\\
\;\;\;\;\begin{array}{l}
t_1 := \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\\
\left(\left(\left|t_1\right| \cdot \sqrt{t_1}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{d}}}{\ell}}\right)\right)\right) \cdot \left(1 - t_0 \cdot \frac{h}{\ell}\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 - \frac{h \cdot t_0}{\ell}\right) \cdot \frac{d}{\left|\sqrt[3]{h}\right|}}{\sqrt{\sqrt[3]{h}} \cdot \sqrt{\ell}}\\


\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
 (let* ((t_0 (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))))
   (if (<= d 2.1521460601640103e-293)
     (let* ((t_1 (/ (cbrt d) (cbrt h))))
       (*
        (*
         (* (fabs t_1) (sqrt t_1))
         (*
          (fabs (cbrt d))
          (*
           (sqrt (cbrt (* (cbrt d) (cbrt d))))
           (sqrt (/ (cbrt (cbrt d)) l)))))
        (- 1.0 (* t_0 (/ h l)))))
     (/
      (* (- 1.0 (/ (* h t_0) l)) (/ d (fabs (cbrt h))))
      (* (sqrt (cbrt h)) (sqrt 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 t_0 = 0.5 * pow(((M * D) / (d * 2.0)), 2.0);
	double tmp;
	if (d <= 2.1521460601640103e-293) {
		double t_1_1 = cbrt(d) / cbrt(h);
		tmp = ((fabs(t_1_1) * sqrt(t_1_1)) * (fabs(cbrt(d)) * (sqrt(cbrt(cbrt(d) * cbrt(d))) * sqrt(cbrt(cbrt(d)) / l)))) * (1.0 - (t_0 * (h / l)));
	} else {
		tmp = ((1.0 - ((h * t_0) / l)) * (d / fabs(cbrt(h)))) / (sqrt(cbrt(h)) * sqrt(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 d < 2.1521460601640103e-293

    1. Initial program 26.8

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

    2. Simplified26.8

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

    4. Applied add-cube-cbrt_binary6427.0

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

    5. Applied add-cube-cbrt_binary6427.1

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

    6. Applied times-frac_binary6427.1

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

    7. Applied sqrt-prod_binary6421.7

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

    8. Simplified21.0

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

    10. Applied *-un-lft-identity_binary6421.0

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

    11. Applied add-cube-cbrt_binary6421.3

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

    12. Applied times-frac_binary6421.3

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

    13. Applied sqrt-prod_binary6417.5

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

    14. Simplified17.5

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

    16. Applied *-un-lft-identity_binary6417.5

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

    17. Applied add-cube-cbrt_binary6417.5

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

    18. Applied cbrt-prod_binary6417.5

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

    19. Applied times-frac_binary6417.5

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

    20. Applied sqrt-prod_binary6417.0

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

    if 2.1521460601640103e-293 < d

    1. Initial program 25.7

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

    2. Simplified25.7

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

    4. Applied add-cube-cbrt_binary6426.0

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

    5. Applied add-cube-cbrt_binary6426.1

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

    6. Applied times-frac_binary6426.1

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

    7. Applied sqrt-prod_binary6420.2

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

    8. Simplified19.5

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

    10. Applied *-un-lft-identity_binary6419.5

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

    11. Applied add-cube-cbrt_binary6419.7

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

    12. Applied times-frac_binary6419.7

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

    13. Applied sqrt-prod_binary6416.5

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

    14. Simplified16.5

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

    16. Applied associate-*r/_binary6414.2

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

    17. Simplified14.2

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

    19. Applied sqrt-div_binary6413.0

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

    20. Applied associate-*r/_binary6413.0

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

    21. Applied sqrt-div_binary6413.0

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

    22. Applied associate-*r/_binary6413.0

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

    23. Applied frac-times_binary6414.2

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

    24. Applied associate-*l/_binary6414.2

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

    25. Simplified13.7

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.1521460601640103 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{d}}}{\ell}}\right)\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}\right) \cdot \frac{d}{\left|\sqrt[3]{h}\right|}}{\sqrt{\sqrt[3]{h}} \cdot \sqrt{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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)))))