Toniolo and Linder, Equation (13)

Average Error: 33.2 → 25.5

Time: 1.0m

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}\;t \le -2.2929690545772815 \cdot 10^{+195}:\\ \;\;\;\;{\left(\left(U \cdot \left(n \cdot t + \frac{\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell - \frac{U - U*}{\frac{Om}{\ell \cdot n}}\right)}{Om}\right)\right) \cdot 2\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 4.7931236222139514 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2 + \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \cdot \left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)}\\ \mathbf{elif}\;t \le 9.503639693249915 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(-\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t \le 8.989646100831337 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\\ \mathbf{elif}\;t \le 2.3441672195036816 \cdot 10^{+280}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(-\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r7421791 = 2.0;
        double r7421792 = n;
        double r7421793 = r7421791 * r7421792;
        double r7421794 = U;
        double r7421795 = r7421793 * r7421794;
        double r7421796 = t;
        double r7421797 = l;
        double r7421798 = r7421797 * r7421797;
        double r7421799 = Om;
        double r7421800 = r7421798 / r7421799;
        double r7421801 = r7421791 * r7421800;
        double r7421802 = r7421796 - r7421801;
        double r7421803 = r7421797 / r7421799;
        double r7421804 = pow(r7421803, r7421791);
        double r7421805 = r7421792 * r7421804;
        double r7421806 = U_;
        double r7421807 = r7421794 - r7421806;
        double r7421808 = r7421805 * r7421807;
        double r7421809 = r7421802 - r7421808;
        double r7421810 = r7421795 * r7421809;
        double r7421811 = sqrt(r7421810);
        return r7421811;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r7421812 = t;
        double r7421813 = -2.2929690545772815e+195;
        bool r7421814 = r7421812 <= r7421813;
        double r7421815 = U;
        double r7421816 = n;
        double r7421817 = r7421816 * r7421812;
        double r7421818 = l;
        double r7421819 = r7421818 * r7421816;
        double r7421820 = -2.0;
        double r7421821 = r7421820 * r7421818;
        double r7421822 = U_;
        double r7421823 = r7421815 - r7421822;
        double r7421824 = Om;
        double r7421825 = r7421824 / r7421819;
        double r7421826 = r7421823 / r7421825;
        double r7421827 = r7421821 - r7421826;
        double r7421828 = r7421819 * r7421827;
        double r7421829 = r7421828 / r7421824;
        double r7421830 = r7421817 + r7421829;
        double r7421831 = r7421815 * r7421830;
        double r7421832 = 2.0;
        double r7421833 = r7421831 * r7421832;
        double r7421834 = 0.5;
        double r7421835 = pow(r7421833, r7421834);
        double r7421836 = 4.7931236222139514e-300;
        bool r7421837 = r7421812 <= r7421836;
        double r7421838 = r7421818 / r7421824;
        double r7421839 = r7421838 * r7421816;
        double r7421840 = r7421839 * r7421815;
        double r7421841 = r7421839 * r7421823;
        double r7421842 = r7421821 - r7421841;
        double r7421843 = r7421840 * r7421842;
        double r7421844 = r7421843 * r7421832;
        double r7421845 = r7421832 * r7421816;
        double r7421846 = r7421845 * r7421815;
        double r7421847 = r7421846 * r7421812;
        double r7421848 = cbrt(r7421847);
        double r7421849 = r7421848 * r7421848;
        double r7421850 = r7421848 * r7421849;
        double r7421851 = r7421844 + r7421850;
        double r7421852 = sqrt(r7421851);
        double r7421853 = 9.503639693249915e-246;
        bool r7421854 = r7421812 <= r7421853;
        double r7421855 = r7421815 * r7421816;
        double r7421856 = r7421832 * r7421855;
        double r7421857 = sqrt(r7421856);
        double r7421858 = r7421832 * r7421818;
        double r7421859 = -r7421839;
        double r7421860 = r7421859 * r7421823;
        double r7421861 = r7421858 - r7421860;
        double r7421862 = r7421838 * r7421861;
        double r7421863 = r7421812 - r7421862;
        double r7421864 = sqrt(r7421863);
        double r7421865 = r7421857 * r7421864;
        double r7421866 = 8.989646100831337e+233;
        bool r7421867 = r7421812 <= r7421866;
        double r7421868 = r7421815 * r7421812;
        double r7421869 = r7421868 * r7421845;
        double r7421870 = r7421869 + r7421844;
        double r7421871 = sqrt(r7421870);
        double r7421872 = sqrt(r7421871);
        double r7421873 = r7421872 * r7421872;
        double r7421874 = 2.3441672195036816e+280;
        bool r7421875 = r7421812 <= r7421874;
        double r7421876 = r7421875 ? r7421865 : r7421873;
        double r7421877 = r7421867 ? r7421873 : r7421876;
        double r7421878 = r7421854 ? r7421865 : r7421877;
        double r7421879 = r7421837 ? r7421852 : r7421878;
        double r7421880 = r7421814 ? r7421835 : r7421879;
        return r7421880;
}

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 t < -2.2929690545772815e+195

   1. Initial program 36.9

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

   3. Applied *-un-lft-identity 36.9

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

   4. Applied associate-*r* 36.9

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

   5. Simplified 33.5

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

   7. Applied sub-neg 33.5

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

   8. Applied distribute-rgt-in 33.5

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

   9. Simplified 29.7

      \[\leadsto \sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \color{blue}{\left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\]
   10. Using strategy rm

   11. Applied pow1 29.7

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

   12. Applied pow1 29.7

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

   13. Applied pow1 29.7

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

   14. Applied pow1 29.7

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

   15. Applied pow-prod-down 29.7

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

   16. Applied pow-prod-down 29.7

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

   17. Applied pow-prod-down 29.7

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

   18. Applied pow1 29.7

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

   19. Applied pow-prod-down 29.7

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

   20. Simplified 29.7

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

   22. Applied pow1 29.7

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

   23. Applied sqrt-pow1 29.7

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

   24. Simplified 29.9

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

   if -2.2929690545772815e+195 < t < 4.7931236222139514e-300

   1. Initial program 32.1

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

   3. Applied *-un-lft-identity 32.1

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

   4. Applied associate-*r* 32.1

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

   5. Simplified 27.9

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

   7. Applied sub-neg 27.9

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

   8. Applied distribute-rgt-in 27.9

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

   9. Simplified 24.0

      \[\leadsto \sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \color{blue}{\left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 24.3

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

   if 4.7931236222139514e-300 < t < 9.503639693249915e-246 or 8.989646100831337e+233 < t < 2.3441672195036816e+280

   1. Initial program 36.6

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

   3. Applied *-un-lft-identity 36.6

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

   4. Applied associate-*r* 36.6

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

   5. Simplified 31.9

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

   7. Applied sqrt-prod 27.1

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

   8. Simplified 27.1

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

   if 9.503639693249915e-246 < t < 8.989646100831337e+233 or 2.3441672195036816e+280 < t

   1. Initial program 32.7

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

   3. Applied *-un-lft-identity 32.7

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

   4. Applied associate-*r* 32.7

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

   5. Simplified 29.0

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

   7. Applied sub-neg 29.0

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

   8. Applied distribute-rgt-in 29.0

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

   9. Simplified 25.1

      \[\leadsto \sqrt{t \cdot \left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot 1\right) + \color{blue}{\left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\]
   10. Using strategy rm

   11. Applied pow1 25.1

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

   12. Applied pow1 25.1

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

   13. Applied pow1 25.1

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

   14. Applied pow1 25.1

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

   15. Applied pow-prod-down 25.1

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

   16. Applied pow-prod-down 25.1

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

   17. Applied pow-prod-down 25.1

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

   18. Applied pow1 25.1

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

   19. Applied pow-prod-down 25.1

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

   20. Simplified 25.3

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

   22. Applied add-sqr-sqrt 25.3

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

   23. Applied sqrt-prod 25.5

       \[\leadsto \color{blue}{\sqrt{\sqrt{{\left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right)}^{1} + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{{\left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right)}^{1} + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 25.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.2929690545772815 \cdot 10^{+195}:\\ \;\;\;\;{\left(\left(U \cdot \left(n \cdot t + \frac{\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell - \frac{U - U*}{\frac{Om}{\ell \cdot n}}\right)}{Om}\right)\right) \cdot 2\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 4.7931236222139514 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2 + \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \cdot \left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)}\\ \mathbf{elif}\;t \le 9.503639693249915 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(-\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t \le 8.989646100831337 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\\ \mathbf{elif}\;t \le 2.3441672195036816 \cdot 10^{+280}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(-\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))