Average Error: 26.1 → 14.4
Time: 14.8s
Precision: binary64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \leq -1.9552656861105664 \cdot 10^{-304}:\\ \;\;\;\;\left(\left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M}{\frac{d}{\frac{D}{2}}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 4.693898141863818 \cdot 10^{-15}:\\ \;\;\;\;\frac{d \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 + \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}}\right)\right) \cdot \left(1 - \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}}\right)\\ \end{array}\]
\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}\;d \leq -1.9552656861105664 \cdot 10^{-304}:\\
\;\;\;\;\left(\left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M}{\frac{d}{\frac{D}{2}}}\right)}^{2}\right)}{\ell}\right)\\

\mathbf{elif}\;d \leq 4.693898141863818 \cdot 10^{-15}:\\
\;\;\;\;\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 \sqrt{\ell}}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 + \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}}\right)\right) \cdot \left(1 - \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\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 (<= d -1.9552656861105664e-304)
   (*
    (*
     (* (fabs (/ 1.0 (cbrt h))) (sqrt (/ d (cbrt h))))
     (* (fabs (/ (cbrt d) (cbrt l))) (sqrt (/ (cbrt d) (cbrt l)))))
    (- 1.0 (/ (* h (* 0.5 (pow (/ M (/ d (/ D 2.0))) 2.0))) l)))
   (if (<= d 4.693898141863818e-15)
     (/
      (* d (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l))))
      (* (sqrt h) (sqrt l)))
     (*
      (*
       (*
        (* (fabs (/ 1.0 (cbrt h))) (sqrt (/ d (cbrt h))))
        (* (fabs (/ (cbrt d) (cbrt l))) (sqrt (/ (cbrt d) (cbrt l)))))
       (+ 1.0 (sqrt (/ (* h (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))) l))))
      (- 1.0 (sqrt (/ (* h (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))) l)))))))
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 ((d <= -1.9552656861105664e-304)) {
		tmp = ((double) (((double) (((double) (((double) fabs((1.0 / ((double) cbrt(h))))) * ((double) sqrt((d / ((double) cbrt(h))))))) * ((double) (((double) fabs((((double) cbrt(d)) / ((double) cbrt(l))))) * ((double) sqrt((((double) cbrt(d)) / ((double) cbrt(l))))))))) * ((double) (1.0 - (((double) (h * ((double) (0.5 * ((double) pow((M / (d / (D / 2.0))), 2.0)))))) / l)))));
	} else {
		double tmp_1;
		if ((d <= 4.693898141863818e-15)) {
			tmp_1 = (((double) (d * ((double) (1.0 - ((double) (((double) (0.5 * ((double) pow((((double) (M * D)) / ((double) (d * 2.0))), 2.0)))) * (h / l))))))) / ((double) (((double) sqrt(h)) * ((double) sqrt(l)))));
		} else {
			tmp_1 = ((double) (((double) (((double) (((double) (((double) fabs((1.0 / ((double) cbrt(h))))) * ((double) sqrt((d / ((double) cbrt(h))))))) * ((double) (((double) fabs((((double) cbrt(d)) / ((double) cbrt(l))))) * ((double) sqrt((((double) cbrt(d)) / ((double) cbrt(l))))))))) * ((double) (1.0 + ((double) sqrt((((double) (h * ((double) (0.5 * ((double) pow((((double) (M * D)) / ((double) (d * 2.0))), 2.0)))))) / l))))))) * ((double) (1.0 - ((double) sqrt((((double) (h * ((double) (0.5 * ((double) pow((((double) (M * D)) / ((double) (d * 2.0))), 2.0)))))) / l)))))));
		}
		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 3 regimes

  2. if d < -1.9552656861105664e-304

    1. Initial program 26.0

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

    2. Simplified26.0

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

    4. Applied add-cube-cbrt_binary6426.2

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

    5. Applied add-cube-cbrt_binary6426.3

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

    6. Applied times-frac_binary6426.3

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

    7. Applied sqrt-prod_binary6422.0

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

    8. Simplified21.8

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

    10. Applied add-cube-cbrt_binary6421.9

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

    11. Applied *-un-lft-identity_binary6421.9

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

    12. Applied times-frac_binary6421.9

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

    13. Applied sqrt-prod_binary6416.8

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

    14. Simplified16.8

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

    16. Applied associate-*r/_binary6413.8

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

    18. Applied associate-/l*_binary6414.3

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

    19. Simplified14.3

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

    if -1.9552656861105664e-304 < d < 4.693898141863818e-15

    1. Initial program 31.0

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

    2. Simplified31.0

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

    4. Applied sqrt-div_binary6425.4

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

    5. Applied sqrt-div_binary6423.9

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

    6. Applied frac-times_binary6423.9

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

    7. Applied associate-*l/_binary6422.2

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

    8. Simplified22.1

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

    if 4.693898141863818e-15 < d

    1. Initial program 22.4

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

    2. Simplified22.4

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

    4. Applied add-cube-cbrt_binary6422.7

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

    5. Applied add-cube-cbrt_binary6422.8

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

    6. Applied times-frac_binary6422.8

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

    7. Applied sqrt-prod_binary6419.8

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

    8. Simplified19.8

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

    10. Applied add-cube-cbrt_binary6419.9

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

    11. Applied *-un-lft-identity_binary6419.9

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

    12. Applied times-frac_binary6419.9

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

    13. Applied sqrt-prod_binary6412.2

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

    14. Simplified12.2

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

    16. Applied associate-*r/_binary648.1

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

    18. Applied add-sqr-sqrt_binary648.1

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

    19. Applied *-un-lft-identity_binary648.1

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

    20. Applied difference-of-squares_binary648.1

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

    21. Applied associate-*r*_binary648.1

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

    22. Simplified8.1

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

  4. Final simplification14.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9552656861105664 \cdot 10^{-304}:\\ \;\;\;\;\left(\left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{h \cdot \left(0.5 \cdot {\left(\frac{M}{\frac{d}{\frac{D}{2}}}\right)}^{2}\right)}{\ell}\right)\\ \mathbf{elif}\;d \leq 4.693898141863818 \cdot 10^{-15}:\\ \;\;\;\;\frac{d \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}{\sqrt{h} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 + \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}}\right)\right) \cdot \left(1 - \sqrt{\frac{h \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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)))))