Toniolo and Linder, Equation (13)

Average Error: 33.6 → 28.7

Time: 3.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -2.2322358565558638 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}\right)\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}}\\ \mathbf{elif}\;U \le -1.2906716725444144 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot \left(t - \left(\ell \cdot 2 + \left(U - U*\right) \cdot \frac{n \cdot \ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;U \le 1.406881514153623 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{U - U*}{\frac{Om}{\frac{\ell}{\frac{Om}{\ell}} \cdot n}}\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \cdot \left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r25071808 = 2.0;
        double r25071809 = n;
        double r25071810 = r25071808 * r25071809;
        double r25071811 = U;
        double r25071812 = r25071810 * r25071811;
        double r25071813 = t;
        double r25071814 = l;
        double r25071815 = r25071814 * r25071814;
        double r25071816 = Om;
        double r25071817 = r25071815 / r25071816;
        double r25071818 = r25071808 * r25071817;
        double r25071819 = r25071813 - r25071818;
        double r25071820 = r25071814 / r25071816;
        double r25071821 = pow(r25071820, r25071808);
        double r25071822 = r25071809 * r25071821;
        double r25071823 = U_;
        double r25071824 = r25071811 - r25071823;
        double r25071825 = r25071822 * r25071824;
        double r25071826 = r25071819 - r25071825;
        double r25071827 = r25071812 * r25071826;
        double r25071828 = sqrt(r25071827);
        return r25071828;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r25071829 = U;
        double r25071830 = -2.2322358565558638e-129;
        bool r25071831 = r25071829 <= r25071830;
        double r25071832 = 2.0;
        double r25071833 = n;
        double r25071834 = r25071832 * r25071833;
        double r25071835 = r25071834 * r25071829;
        double r25071836 = t;
        double r25071837 = l;
        double r25071838 = Om;
        double r25071839 = r25071838 / r25071837;
        double r25071840 = r25071837 / r25071839;
        double r25071841 = r25071832 * r25071840;
        double r25071842 = r25071836 - r25071841;
        double r25071843 = r25071837 / r25071838;
        double r25071844 = U_;
        double r25071845 = r25071829 - r25071844;
        double r25071846 = r25071843 * r25071833;
        double r25071847 = r25071845 * r25071846;
        double r25071848 = r25071843 * r25071847;
        double r25071849 = r25071842 - r25071848;
        double r25071850 = cbrt(r25071849);
        double r25071851 = r25071850 * r25071850;
        double r25071852 = r25071835 * r25071851;
        double r25071853 = r25071852 * r25071850;
        double r25071854 = sqrt(r25071853);
        double r25071855 = -1.2906716725444144e-209;
        bool r25071856 = r25071829 <= r25071855;
        double r25071857 = r25071837 * r25071832;
        double r25071858 = r25071833 * r25071837;
        double r25071859 = r25071858 / r25071838;
        double r25071860 = r25071845 * r25071859;
        double r25071861 = r25071857 + r25071860;
        double r25071862 = r25071861 * r25071843;
        double r25071863 = r25071836 - r25071862;
        double r25071864 = r25071833 * r25071863;
        double r25071865 = r25071864 * r25071829;
        double r25071866 = r25071865 * r25071832;
        double r25071867 = sqrt(r25071866);
        double r25071868 = 1.406881514153623e-22;
        bool r25071869 = r25071829 <= r25071868;
        double r25071870 = r25071840 * r25071833;
        double r25071871 = r25071838 / r25071870;
        double r25071872 = r25071845 / r25071871;
        double r25071873 = r25071842 - r25071872;
        double r25071874 = r25071873 * r25071829;
        double r25071875 = r25071834 * r25071874;
        double r25071876 = sqrt(r25071875);
        double r25071877 = r25071835 * r25071849;
        double r25071878 = cbrt(r25071877);
        double r25071879 = r25071878 * r25071878;
        double r25071880 = r25071878 * r25071879;
        double r25071881 = sqrt(r25071880);
        double r25071882 = r25071869 ? r25071876 : r25071881;
        double r25071883 = r25071856 ? r25071867 : r25071882;
        double r25071884 = r25071831 ? r25071854 : r25071883;
        return r25071884;
}

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 U < -2.2322358565558638e-129

   1. Initial program 28.9

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

   3. Applied associate-/l* 26.0

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

   5. Applied pow1 26.0

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

   6. Applied pow1 26.0

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

   7. Applied pow-prod-down 26.0

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

   8. Simplified 25.0

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

   10. Applied add-cube-cbrt 25.3

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

   11. Applied associate-*r* 25.3

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

   if -2.2322358565558638e-129 < U < -1.2906716725444144e-209

   1. Initial program 37.8

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

   3. Applied associate-/l* 36.1

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

   5. Applied pow1 36.1

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

   6. Applied pow1 36.1

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

   7. Applied pow-prod-down 36.1

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

   8. Simplified 34.6

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

   10. Applied pow1 34.6

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

   11. Applied pow1 34.6

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

   12. Applied pow1 34.6

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

   13. Applied pow-prod-down 34.6

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

   14. Applied pow-prod-down 34.6

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

   15. Simplified 33.0

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

   if -1.2906716725444144e-209 < U < 1.406881514153623e-22

   1. Initial program 38.8

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

   3. Applied associate-/l* 36.7

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

   5. Applied *-un-lft-identity 36.7

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

   6. Applied associate-*r* 36.7

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

   7. Simplified 36.1

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

   9. Applied associate-*l* 31.0

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

   10. Simplified 32.1

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

   if 1.406881514153623e-22 < U

   1. Initial program 28.8

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

   3. Applied associate-/l* 25.8

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

   5. Applied pow1 25.8

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

   6. Applied pow1 25.8

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

   7. Applied pow-prod-down 25.8

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

   8. Simplified 24.8

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

   10. Applied add-cube-cbrt 25.1

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

4. Final simplification 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -2.2322358565558638 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}\right)\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)}}\\ \mathbf{elif}\;U \le -1.2906716725444144 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot \left(t - \left(\ell \cdot 2 + \left(U - U*\right) \cdot \frac{n \cdot \ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U\right) \cdot 2}\\ \mathbf{elif}\;U \le 1.406881514153623 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{U - U*}{\frac{Om}{\frac{\ell}{\frac{Om}{\ell}} \cdot n}}\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \cdot \left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \frac{\ell}{Om} \cdot \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(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*))))))