Toniolo and Linder, Equation (13)

Average Error: 34.4 → 31.2

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -9.014626965436397950093167243230168496464 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U* - U\right) \cdot n\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 1.008725107202353820896882228886919841361 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{\left(\left(\left(\sqrt[3]{U* - U} \cdot \left(\sqrt[3]{U* - U} \cdot \left(\sqrt[3]{U* - U} \cdot 0\right)\right) - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) + t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;n \le 31641216734603.953125:\\ \;\;\;\;\sqrt{\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - t\right) \cdot \left(\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(-2 \cdot n\right)\right) + \left(\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{2} \cdot n\right) \cdot {\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{2}\right) \cdot \left(\left(U* - U\right) \cdot \left(\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right)\right)\right) \cdot \sqrt[3]{U}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot {\left(\sqrt[3]{U* - U}\right)}^{3} - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell - t\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r194832 = 2.0;
        double r194833 = n;
        double r194834 = r194832 * r194833;
        double r194835 = U;
        double r194836 = r194834 * r194835;
        double r194837 = t;
        double r194838 = l;
        double r194839 = r194838 * r194838;
        double r194840 = Om;
        double r194841 = r194839 / r194840;
        double r194842 = r194832 * r194841;
        double r194843 = r194837 - r194842;
        double r194844 = r194838 / r194840;
        double r194845 = pow(r194844, r194832);
        double r194846 = r194833 * r194845;
        double r194847 = U_;
        double r194848 = r194835 - r194847;
        double r194849 = r194846 * r194848;
        double r194850 = r194843 - r194849;
        double r194851 = r194836 * r194850;
        double r194852 = sqrt(r194851);
        return r194852;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r194853 = n;
        double r194854 = -9.014626965436398e-81;
        bool r194855 = r194853 <= r194854;
        double r194856 = 2.0;
        double r194857 = r194856 * r194853;
        double r194858 = l;
        double r194859 = Om;
        double r194860 = r194858 / r194859;
        double r194861 = pow(r194860, r194856);
        double r194862 = U_;
        double r194863 = U;
        double r194864 = r194862 - r194863;
        double r194865 = r194864 * r194853;
        double r194866 = r194861 * r194865;
        double r194867 = r194860 * r194858;
        double r194868 = r194867 * r194856;
        double r194869 = t;
        double r194870 = r194868 - r194869;
        double r194871 = r194866 - r194870;
        double r194872 = r194871 * r194863;
        double r194873 = r194857 * r194872;
        double r194874 = sqrt(r194873);
        double r194875 = 1.0087251072023538e-199;
        bool r194876 = r194853 <= r194875;
        double r194877 = cbrt(r194864);
        double r194878 = 0.0;
        double r194879 = r194877 * r194878;
        double r194880 = r194877 * r194879;
        double r194881 = r194877 * r194880;
        double r194882 = r194856 * r194858;
        double r194883 = r194882 * r194860;
        double r194884 = r194881 - r194883;
        double r194885 = r194884 + r194869;
        double r194886 = r194885 * r194857;
        double r194887 = r194886 * r194863;
        double r194888 = sqrt(r194887);
        double r194889 = 31641216734603.953;
        bool r194890 = r194853 <= r194889;
        double r194891 = r194883 - r194869;
        double r194892 = cbrt(r194863);
        double r194893 = r194892 * r194892;
        double r194894 = -r194857;
        double r194895 = r194893 * r194894;
        double r194896 = r194891 * r194895;
        double r194897 = cbrt(r194858);
        double r194898 = cbrt(r194859);
        double r194899 = r194897 / r194898;
        double r194900 = pow(r194899, r194856);
        double r194901 = r194900 * r194853;
        double r194902 = r194897 * r194897;
        double r194903 = r194898 * r194898;
        double r194904 = r194902 / r194903;
        double r194905 = pow(r194904, r194856);
        double r194906 = r194901 * r194905;
        double r194907 = r194857 * r194893;
        double r194908 = r194864 * r194907;
        double r194909 = r194906 * r194908;
        double r194910 = r194896 + r194909;
        double r194911 = r194910 * r194892;
        double r194912 = sqrt(r194911);
        double r194913 = r194863 * r194857;
        double r194914 = r194861 * r194853;
        double r194915 = 3.0;
        double r194916 = pow(r194877, r194915);
        double r194917 = r194914 * r194916;
        double r194918 = r194856 * r194860;
        double r194919 = r194918 * r194858;
        double r194920 = r194919 - r194869;
        double r194921 = r194917 - r194920;
        double r194922 = r194913 * r194921;
        double r194923 = sqrt(r194922);
        double r194924 = r194890 ? r194912 : r194923;
        double r194925 = r194876 ? r194888 : r194924;
        double r194926 = r194855 ? r194874 : r194925;
        return r194926;
}

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 < -9.014626965436398e-81

   1. Initial program 33.2

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

   2. Simplified 34.0

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

   4. Applied associate-*l* 31.1

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

   5. Simplified 35.2

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

   if -9.014626965436398e-81 < n < 1.0087251072023538e-199

   1. Initial program 37.1

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

   2. Simplified 30.3

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

   4. Applied add-cube-cbrt 30.3

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

   5. Applied associate-*r* 30.3

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

   6. Simplified 30.3

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

   7. Taylor expanded around 0 30.9

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \left(\left(\sqrt[3]{U* - U} \cdot \left(\sqrt[3]{U* - U} \cdot \color{blue}{0}\right)\right) \cdot \sqrt[3]{U* - U} - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U}\]

   if 1.0087251072023538e-199 < n < 31641216734603.953

   1. Initial program 31.7

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

   2. Simplified 26.0

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

   4. Applied add-cube-cbrt 26.0

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

   5. Applied associate-*r* 26.0

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

   6. Simplified 26.0

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

   8. Applied add-cube-cbrt 26.4

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

   9. Applied associate-*r* 26.4

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

   10. Simplified 26.5

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

   12. Applied sub-neg 26.5

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

   13. Applied distribute-lft-in 26.5

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

   14. Simplified 27.6

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

   15. Simplified 27.6

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

   17. Applied add-cube-cbrt 27.7

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

   18. Applied add-cube-cbrt 27.7

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

   19. Applied times-frac 27.7

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

   20. Applied unpow-prod-down 27.7

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

   21. Applied associate-*l* 26.7

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

   22. Simplified 26.7

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

   if 31641216734603.953 < n

   1. Initial program 33.8

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

   2. Simplified 36.8

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

   4. Applied add-cube-cbrt 36.8

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

   5. Applied associate-*r* 36.8

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

   6. Simplified 36.8

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

   8. Applied pow1 36.8

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

   9. Applied pow1 36.8

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

   10. Applied pow1 36.8

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

   11. Applied pow1 36.8

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

   12. Applied pow-prod-down 36.8

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

   13. Applied pow-prod-down 36.8

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

   14. Applied pow-prod-down 36.8

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

   15. Simplified 31.0

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

4. Final simplification 31.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -9.014626965436397950093167243230168496464 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U* - U\right) \cdot n\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 1.008725107202353820896882228886919841361 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{\left(\left(\left(\sqrt[3]{U* - U} \cdot \left(\sqrt[3]{U* - U} \cdot \left(\sqrt[3]{U* - U} \cdot 0\right)\right) - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right) + t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;n \le 31641216734603.953125:\\ \;\;\;\;\sqrt{\left(\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - t\right) \cdot \left(\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(-2 \cdot n\right)\right) + \left(\left({\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{2} \cdot n\right) \cdot {\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{2}\right) \cdot \left(\left(U* - U\right) \cdot \left(\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right)\right)\right) \cdot \sqrt[3]{U}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot {\left(\sqrt[3]{U* - U}\right)}^{3} - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell - t\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))