Toniolo and Linder, Equation (13)

Average Error: 34.3 → 27.4

Time: 37.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le 4.437888771927319 \cdot 10^{-300}:\\ \;\;\;\;\left|\sqrt[3]{\left(2 \cdot n\right) \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\left(2 \cdot n\right) \cdot U} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot U}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\left(2 \cdot n\right) \cdot U}} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\\ \mathbf{elif}\;n \le 1.258070492458926 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \le 2.71812111302340703 \cdot 10^{262}:\\ \;\;\;\;\left|\sqrt[3]{\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)} \cdot \sqrt[3]{\sqrt[3]{U}}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\left(2 \cdot n\right) \cdot U} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot U}}} \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{U}}} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

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 ((n <= 4.437888771927319e-300)) {
		VAR = ((double) (((double) fabs(((double) cbrt(((double) (((double) (2.0 * n)) * U)))))) * ((double) (((double) sqrt(((double) cbrt(((double) (((double) cbrt(((double) (((double) (2.0 * n)) * U)))) * ((double) cbrt(((double) (((double) (2.0 * n)) * U)))))))))) * ((double) sqrt(((double) (((double) cbrt(((double) cbrt(((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_))))))))))))));
	} else {
		double VAR_1;
		if ((n <= 1.258070492458926e-112)) {
			VAR_1 = ((double) (((double) sqrt(n)) * ((double) sqrt(((double) (((double) (2.0 * 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_))))))))))));
		} else {
			double VAR_2;
			if ((n <= 2.718121113023407e+262)) {
				VAR_2 = ((double) (((double) fabs(((double) (((double) cbrt(((double) (((double) (2.0 * n)) * ((double) (((double) cbrt(U)) * ((double) cbrt(U)))))))) * ((double) cbrt(((double) cbrt(U)))))))) * ((double) (((double) sqrt(((double) cbrt(((double) (((double) cbrt(((double) (((double) (2.0 * n)) * U)))) * ((double) cbrt(((double) (((double) (2.0 * n)) * U)))))))))) * ((double) (((double) sqrt(((double) cbrt(((double) cbrt(((double) (((double) (2.0 * n)) * ((double) (((double) cbrt(U)) * ((double) cbrt(U)))))))))))) * ((double) sqrt(((double) (((double) cbrt(((double) cbrt(((double) cbrt(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_))))))))))))))));
			} else {
				VAR_2 = ((double) (((double) sqrt(((double) (2.0 * n)))) * ((double) sqrt(((double) (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_))))))))))));
			}
			VAR_1 = VAR_2;
		}
		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 4 regimes

2. if n < 4.437888771927319e-300

   1. Initial program 35.1

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

   3. Applied add-cube-cbrt 35.4

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

   4. Applied associate-*l* 35.4

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

   5. Applied sqrt-prod 30.9

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

   6. Simplified 30.9

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

   8. Applied add-cube-cbrt 31.0

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

   9. Applied cbrt-prod 31.0

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

   10. Applied associate-*l* 31.0

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

   11. Applied sqrt-prod 29.6

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

   if 4.437888771927319e-300 < n < 1.258070492458926e-112

   1. Initial program 37.0

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

   3. Applied *-commutative 37.0

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

   4. Applied associate-*l* 37.0

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

   5. Applied associate-*l* 37.0

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

   6. Applied sqrt-prod 28.6

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

   if 1.258070492458926e-112 < n < 2.718121113023407e+262

   1. Initial program 30.8

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

   3. Applied add-cube-cbrt 31.1

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

   4. Applied associate-*l* 31.1

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

   5. Applied sqrt-prod 25.8

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

   6. Simplified 25.8

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

   8. Applied add-cube-cbrt 25.8

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

   9. Applied cbrt-prod 25.8

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

   10. Applied associate-*l* 25.9

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

   11. Applied sqrt-prod 23.8

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

   13. Applied add-cube-cbrt 23.8

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

   14. Applied associate-*r* 23.8

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

   15. Applied cbrt-prod 23.9

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

   17. Applied add-cube-cbrt 23.9

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

   18. Applied associate-*r* 23.9

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

   19. Applied cbrt-prod 24.0

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

   20. Applied cbrt-prod 24.0

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

   21. Applied associate-*l* 24.0

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

   22. Applied sqrt-prod 23.2

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

   if 2.718121113023407e+262 < n

   1. Initial program 39.9

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

   3. Applied associate-*l* 39.7

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

   4. Applied sqrt-prod 15.6

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 27.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le 4.437888771927319 \cdot 10^{-300}:\\ \;\;\;\;\left|\sqrt[3]{\left(2 \cdot n\right) \cdot U}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\left(2 \cdot n\right) \cdot U} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot U}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\left(2 \cdot n\right) \cdot U}} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\\ \mathbf{elif}\;n \le 1.258070492458926 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\left(2 \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \le 2.71812111302340703 \cdot 10^{262}:\\ \;\;\;\;\left|\sqrt[3]{\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)} \cdot \sqrt[3]{\sqrt[3]{U}}\right| \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\left(2 \cdot n\right) \cdot U} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot U}}} \cdot \left(\sqrt{\sqrt[3]{\sqrt[3]{\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{U}}} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(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*))))))