Toniolo and Linder, Equation (13)

Average Error: 35.0 → 19.8

Time: 2.6m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 2.13436 \cdot 10^{-321} \lor \neg \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 8.66906088821881914 \cdot 10^{109}\right):\\ \;\;\;\;\left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}} \cdot \left(\left(t - 2 \cdot \frac{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}{\frac{Om}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(1, t - 2 \cdot \frac{\ell \cdot \ell}{Om}, -\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \end{array}\]

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double temp;
	if ((((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 2.1343635900342e-321) || !((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))) <= 8.669060888218819e+109))) {
		temp = (fabs(cbrt((2.0 * n))) * (fabs(cbrt((cbrt((2.0 * n)) * U))) * (sqrt(cbrt((cbrt((cbrt((2.0 * n)) * U)) * cbrt((cbrt((2.0 * n)) * U))))) * sqrt((cbrt(cbrt((cbrt((2.0 * n)) * U))) * ((t - (2.0 * (((cbrt(l) * cbrt(l)) * (cbrt(l) * cbrt(l))) / (Om / (cbrt(l) * cbrt(l)))))) - ((n * pow((l / Om), 2.0)) * (U - U_42_))))))));
	} else {
		temp = sqrt(((((2.0 * n) * U) * fma(1.0, (t - (2.0 * ((l * l) / Om))), -((pow((l / Om), (2.0 / 2.0)) * (U - U_42_)) * (n * pow((l / Om), (2.0 / 2.0)))))) + (((2.0 * n) * U) * fma(-(pow((l / Om), (2.0 / 2.0)) * (U - U_42_)), (n * pow((l / Om), (2.0 / 2.0))), ((pow((l / Om), (2.0 / 2.0)) * (U - U_42_)) * (n * pow((l / Om), (2.0 / 2.0))))))));
	}
	return temp;
}

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 2 regimes

2. if (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*)))) < 2.1343635900342e-321 or 8.669060888218819e+109 < (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))

   1. Initial program 50.1

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

   3. Applied add-cube-cbrt 50.2

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

   4. Applied associate-*l* 50.2

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

   5. Applied associate-*l* 48.8

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

   6. Applied sqrt-prod 42.1

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

   7. Simplified 42.1

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

   9. Applied add-cube-cbrt 42.2

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

   10. Applied associate-*l* 42.2

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

   11. Applied sqrt-prod 34.7

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

   12. Simplified 34.7

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

   14. Applied add-cube-cbrt 34.7

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

   15. Applied add-cube-cbrt 34.7

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

   16. Applied swap-sqr 34.7

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

   17. Applied associate-/l* 30.4

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

   19. Applied add-cube-cbrt 30.5

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

   20. Applied cbrt-prod 30.5

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

   21. Applied associate-*l* 30.5

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

   22. Applied sqrt-prod 28.2

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

   if 2.1343635900342e-321 < (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*)))) < 8.669060888218819e+109

   1. Initial program 2.3

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

   3. Applied sqr-pow 2.3

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

   4. Applied associate-*r* 1.5

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

   5. Applied associate-*l* 1.6

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

   6. Applied *-un-lft-identity 1.6

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

   7. Applied prod-diff 1.6

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

   8. Applied distribute-lft-in 1.6

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

4. Final simplification 19.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 2.13436 \cdot 10^{-321} \lor \neg \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 8.66906088821881914 \cdot 10^{109}\right):\\ \;\;\;\;\left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}} \cdot \left(\left(t - 2 \cdot \frac{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}{\frac{Om}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(1, t - 2 \cdot \frac{\ell \cdot \ell}{Om}, -\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) + \left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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*))))))