Henrywood and Agarwal, Equation (12)

Average Error: 26.7 → 13.2

Time: 22.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\sqrt[3]{h}}}\\ t_1 := \frac{M \cdot D}{d \cdot 2}\\ t_2 := {t_1}^{2}\\ t_3 := 1 - \left(0.5 \cdot t_2\right) \cdot \frac{h}{\ell}\\ t_4 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot t_3\\ t_5 := \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\\ t_6 := \sqrt{t_5}\\ t_7 := \left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot t_0\right) \cdot \left(\left|t_5\right| \cdot t_6\right)\\ \mathbf{if}\;t_4 \leq -6.047698773616513 \cdot 10^{+304}:\\ \;\;\;\;t_7 \cdot \left(1 - e^{\left(\left(\log 0.5 + 2 \cdot \log t_1\right) + \log \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) + \log \left(\frac{\sqrt[3]{h}}{\ell}\right)}\right)\\ \mathbf{elif}\;t_4 \leq 2.2117198870015957 \cdot 10^{+206}:\\ \;\;\;\;\frac{t_3 \cdot \left(t_0 \cdot \left(t_6 \cdot \left|\sqrt[3]{d}\right|\right)\right)}{\left|\sqrt[3]{h}\right| \cdot \left|\sqrt[3]{\ell}\right|}\\ \mathbf{else}:\\ \;\;\;\;t_7 \cdot \left(1 - \log \left({\left(\sqrt{e^{t_2}}\right)}^{\left(\frac{h}{\ell}\right)}\right)\right)\\ \end{array} \]

FPCore

(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 (sqrt (/ d (cbrt h))))
        (t_1 (/ (* M D) (* d 2.0)))
        (t_2 (pow t_1 2.0))
        (t_3 (- 1.0 (* (* 0.5 t_2) (/ h l))))
        (t_4 (* (* (pow (/ d h) 0.5) (pow (/ d l) 0.5)) t_3))
        (t_5 (/ (cbrt d) (cbrt l)))
        (t_6 (sqrt t_5))
        (t_7 (* (* (fabs (/ 1.0 (cbrt h))) t_0) (* (fabs t_5) t_6))))
   (if (<= t_4 -6.047698773616513e+304)
     (*
      t_7
      (-
       1.0
       (exp
        (+
         (+ (+ (log 0.5) (* 2.0 (log t_1))) (log (* (cbrt h) (cbrt h))))
         (log (/ (cbrt h) l))))))
     (if (<= t_4 2.2117198870015957e+206)
       (/
        (* t_3 (* t_0 (* t_6 (fabs (cbrt d)))))
        (* (fabs (cbrt h)) (fabs (cbrt l))))
       (* t_7 (- 1.0 (log (pow (sqrt (exp t_2)) (/ h l)))))))))

C

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 = sqrt(d / cbrt(h));
	double t_1 = (M * D) / (d * 2.0);
	double t_2 = pow(t_1, 2.0);
	double t_3 = 1.0 - ((0.5 * t_2) * (h / l));
	double t_4 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * t_3;
	double t_5 = cbrt(d) / cbrt(l);
	double t_6 = sqrt(t_5);
	double t_7 = (fabs(1.0 / cbrt(h)) * t_0) * (fabs(t_5) * t_6);
	double tmp;
	if (t_4 <= -6.047698773616513e+304) {
		tmp = t_7 * (1.0 - exp(((log(0.5) + (2.0 * log(t_1))) + log(cbrt(h) * cbrt(h))) + log(cbrt(h) / l)));
	} else if (t_4 <= 2.2117198870015957e+206) {
		tmp = (t_3 * (t_0 * (t_6 * fabs(cbrt(d))))) / (fabs(cbrt(h)) * fabs(cbrt(l)));
	} else {
		tmp = t_7 * (1.0 - log(pow(sqrt(exp(t_2)), (h / l))));
	}
	return tmp;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -6.04769877361651321e304

   1. Initial program 63.8

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

   3. Applied add-cube-cbrt_binary64 63.8

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

   4. Applied *-un-lft-identity_binary64 63.8

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

   5. Applied times-frac_binary64 63.8

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

   6. Applied unpow-prod-down_binary64 63.6

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

   7. Simplified 63.6

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

   8. Simplified 63.6

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

   10. Applied add-cube-cbrt_binary64 63.6

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

   11. Applied add-cube-cbrt_binary64 63.6

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

   12. Applied times-frac_binary64 63.6

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

   13. Applied unpow-prod-down_binary64 62.5

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

   14. Simplified 62.5

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

   15. Simplified 62.5

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

   17. Applied *-un-lft-identity_binary64 62.5

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

   18. Applied add-cube-cbrt_binary64 62.5

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

   19. Applied times-frac_binary64 62.5

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

   20. Applied associate-*r*_binary64 53.7

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

   21. Simplified 53.7

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

   23. Applied add-exp-log_binary64 54.4

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

   24. Applied add-exp-log_binary64 54.4

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

   25. Applied pow-to-exp_binary64 58.7

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

   26. Applied add-exp-log_binary64 58.7

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

   27. Applied prod-exp_binary64 58.7

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

   28. Applied prod-exp_binary64 54.5

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

   29. Applied prod-exp_binary64 49.8

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

   if -6.04769877361651321e304 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 2.2117198870015957e206

   1. Initial program 7.2

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

   3. Applied add-cube-cbrt_binary64 7.6

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

   4. Applied *-un-lft-identity_binary64 7.6

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

   5. Applied times-frac_binary64 7.6

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

   6. Applied unpow-prod-down_binary64 6.4

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

   7. Simplified 6.4

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

   8. Simplified 6.4

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

   10. Applied add-cube-cbrt_binary64 6.6

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

   11. Applied add-cube-cbrt_binary64 6.7

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

   12. Applied times-frac_binary64 6.7

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

   13. Applied unpow-prod-down_binary64 2.7

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

   14. Simplified 2.4

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

   15. Simplified 2.4

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

   17. Applied fabs-div_binary64 2.4

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

   18. Applied associate-*l/_binary64 2.4

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

   19. Applied fabs-div_binary64 2.4

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

   20. Applied associate-*l/_binary64 2.4

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

   21. Applied frac-times_binary64 2.4

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

   22. Applied associate-*l/_binary64 2.0

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

   23. Simplified 2.0

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

   if 2.2117198870015957e206 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 56.7

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

   3. Applied add-cube-cbrt_binary64 56.7

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

   4. Applied *-un-lft-identity_binary64 56.7

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

   5. Applied times-frac_binary64 56.8

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

   6. Applied unpow-prod-down_binary64 42.1

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

   7. Simplified 42.1

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

   8. Simplified 42.1

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

   10. Applied add-cube-cbrt_binary64 42.2

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

   11. Applied add-cube-cbrt_binary64 42.3

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

   12. Applied times-frac_binary64 42.3

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

   13. Applied unpow-prod-down_binary64 36.8

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

   14. Simplified 36.8

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

   15. Simplified 36.8

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

   17. Applied add-log-exp_binary64 36.8

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

   18. Simplified 26.1

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -6.047698773616513 \cdot 10^{+304}:\\ \;\;\;\;\left(\left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - e^{\left(\left(\log 0.5 + 2 \cdot \log \left(\frac{M \cdot D}{d \cdot 2}\right)\right) + \log \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) + \log \left(\frac{\sqrt[3]{h}}{\ell}\right)}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2.2117198870015957 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left|\sqrt[3]{d}\right|\right)\right)}{\left|\sqrt[3]{h}\right| \cdot \left|\sqrt[3]{\ell}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|\frac{1}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \log \left({\left(\sqrt{e^{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}}}\right)}^{\left(\frac{h}{\ell}\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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)))))