Toniolo and Linder, Equation (13)

Average Error: 34.3 → 20.0

Time: 1.3m

Precision: 64

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -9.18064049695707294 \cdot 10^{136}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\ell \le 2.9005648470872701 \cdot 10^{99}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n}} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \sqrt{\left(\sqrt[3]{n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot 1}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return ((double) sqrt(((double) (((double) (((double) (2.0 * n)) * U)) * ((double) (((double) (t - ((double) (2.0 * ((double) (((double) (l * l)) / Om)))))) - ((double) (((double) (n * ((double) pow(((double) (l / Om)), 2.0)))) * ((double) (U - U_42_))))))))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double VAR;
	if ((l <= -9.180640496957073e+136)) {
		VAR = ((double) sqrt(((double) (((double) (((double) (2.0 * n)) * U)) * ((double) (((double) (t - ((double) (2.0 * ((double) (l * ((double) (l / Om)))))))) - ((double) (((double) (n * ((double) pow(((double) (l / Om)), 2.0)))) * ((double) (U - U_42_))))))))));
	} else {
		double VAR_1;
		if ((l <= 2.90056484708727e+99)) {
			VAR_1 = ((double) (((double) sqrt(((double) (2.0 * ((double) (((double) cbrt(n)) * ((double) cbrt(n)))))))) * ((double) (((double) fabs(((double) cbrt(((double) (((double) cbrt(n)) * U)))))) * ((double) (((double) sqrt(((double) cbrt(((double) cbrt(((double) (((double) cbrt(n)) * ((double) cbrt(n)))))))))) * ((double) sqrt(((double) (((double) cbrt(((double) (((double) cbrt(((double) cbrt(n)))) * U)))) * ((double) (((double) (t - ((double) (2.0 * ((double) (((double) (l * l)) / Om)))))) - ((double) (((double) (n * ((double) pow(((double) (l / Om)), ((double) (2.0 / 2.0)))))) * ((double) (((double) pow(((double) (l / Om)), ((double) (2.0 / 2.0)))) * ((double) (U - U_42_))))))))))))))))));
		} else {
			VAR_1 = ((double) (((double) sqrt(((double) (2.0 * ((double) (((double) cbrt(n)) * ((double) cbrt(n)))))))) * ((double) sqrt(((double) (((double) (((double) cbrt(n)) * U)) * ((double) (((double) (t - ((double) (2.0 * ((double) (((double) (l * 1.0)) / ((double) (Om / l)))))))) - ((double) (((double) (n * ((double) pow(((double) (l / Om)), 2.0)))) * ((double) (U - U_42_))))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 3 regimes

2. if l < -9.180640496957073e+136

   1. Initial program 60.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 60.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied times-frac 48.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Simplified 48.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\color{blue}{\ell} \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   if -9.180640496957073e+136 < l < 2.90056484708727e+99

   1. Initial program 27.2

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 27.6

      \[\leadsto \sqrt{\left(\left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}\right)}\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied associate-*r* 27.6

      \[\leadsto \sqrt{\left(\color{blue}{\left(\left(2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \sqrt[3]{n}\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Applied associate-*l* 27.5

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \left(\sqrt[3]{n} \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   6. Applied associate-*l* 26.6

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \left(\left(\sqrt[3]{n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]

   7. Applied sqrt-prod 21.1

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \sqrt{\left(\sqrt[3]{n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 21.2

      \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \sqrt{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{n} \cdot U} \cdot \sqrt[3]{\sqrt[3]{n} \cdot U}\right) \cdot \sqrt[3]{\sqrt[3]{n} \cdot U}\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   10. Applied associate-*l* 21.2

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\sqrt[3]{n} \cdot U} \cdot \sqrt[3]{\sqrt[3]{n} \cdot U}\right) \cdot \left(\sqrt[3]{\sqrt[3]{n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]

   11. Applied sqrt-prod 14.4

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\sqrt[3]{n} \cdot U} \cdot \sqrt[3]{\sqrt[3]{n} \cdot U}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}\]

   12. Simplified 14.4

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\color{blue}{\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right|} \cdot \sqrt{\sqrt[3]{\sqrt[3]{n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 14.4

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\]

   15. Applied cbrt-prod 14.4

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \sqrt[3]{\sqrt[3]{n}}\right)} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\]

   16. Applied associate-*l* 14.4

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\color{blue}{\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \left(\sqrt[3]{\sqrt[3]{n}} \cdot U\right)}} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\]

   17. Applied cbrt-prod 14.4

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{n}} \cdot U}\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\]

   18. Applied associate-*l* 14.7

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \sqrt{\color{blue}{\sqrt[3]{\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{n}} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\right)\]

   19. Applied sqrt-prod 14.4

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n}} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}\right)\]
   20. Using strategy rm

   21. Applied sqr-pow 14.4

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n}} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)}\right)\right)\]

   22. Applied associate-*r* 12.9

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n}} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)}\right)\right)\]

   23. Applied associate-*l* 12.5

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n}} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)}\right)\right)\]

   if 2.90056484708727e+99 < l

   1. Initial program 54.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 54.9

      \[\leadsto \sqrt{\left(\left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}\right)}\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied associate-*r* 54.9

      \[\leadsto \sqrt{\left(\color{blue}{\left(\left(2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \sqrt[3]{n}\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Applied associate-*l* 54.9

      \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \left(\sqrt[3]{n} \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   6. Applied associate-*l* 54.7

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right) \cdot \left(\left(\sqrt[3]{n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]

   7. Applied sqrt-prod 53.8

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \sqrt{\left(\sqrt[3]{n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 53.8

      \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \sqrt{\left(\sqrt[3]{n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \color{blue}{\left(1 \cdot \ell\right)}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   10. Applied associate-*r* 53.8

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \sqrt{\left(\sqrt[3]{n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\left(\ell \cdot 1\right) \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   11. Applied associate-/l* 41.3

       \[\leadsto \sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \sqrt{\left(\sqrt[3]{n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot 1}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 20.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -9.18064049695707294 \cdot 10^{136}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\ell \le 2.9005648470872701 \cdot 10^{99}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \left(\left|\sqrt[3]{\sqrt[3]{n} \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n} \cdot \sqrt[3]{n}}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{n}} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)} \cdot \sqrt{\left(\sqrt[3]{n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot 1}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))