Toniolo and Linder, Equation (13)

Average Error: 34.9 → 30.0

Time: 27.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -2.037058507986194199807262357510988881335 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\left(n \cdot {\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 8.420291904348501272367972443482728416614 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)} \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r144770 = 2.0;
        double r144771 = n;
        double r144772 = r144770 * r144771;
        double r144773 = U;
        double r144774 = r144772 * r144773;
        double r144775 = t;
        double r144776 = l;
        double r144777 = r144776 * r144776;
        double r144778 = Om;
        double r144779 = r144777 / r144778;
        double r144780 = r144770 * r144779;
        double r144781 = r144775 - r144780;
        double r144782 = r144776 / r144778;
        double r144783 = pow(r144782, r144770);
        double r144784 = r144771 * r144783;
        double r144785 = U_;
        double r144786 = r144773 - r144785;
        double r144787 = r144784 * r144786;
        double r144788 = r144781 - r144787;
        double r144789 = r144774 * r144788;
        double r144790 = sqrt(r144789);
        return r144790;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r144791 = n;
        double r144792 = -2.0370585079861942e-134;
        bool r144793 = r144791 <= r144792;
        double r144794 = t;
        double r144795 = 2.0;
        double r144796 = l;
        double r144797 = Om;
        double r144798 = r144796 / r144797;
        double r144799 = r144796 * r144798;
        double r144800 = cbrt(r144796);
        double r144801 = r144800 * r144800;
        double r144802 = 2.0;
        double r144803 = r144795 / r144802;
        double r144804 = pow(r144801, r144803);
        double r144805 = r144791 * r144804;
        double r144806 = r144800 / r144797;
        double r144807 = pow(r144806, r144803);
        double r144808 = r144805 * r144807;
        double r144809 = pow(r144798, r144803);
        double r144810 = U;
        double r144811 = U_;
        double r144812 = r144810 - r144811;
        double r144813 = r144809 * r144812;
        double r144814 = r144808 * r144813;
        double r144815 = fma(r144795, r144799, r144814);
        double r144816 = r144794 - r144815;
        double r144817 = r144795 * r144791;
        double r144818 = r144817 * r144810;
        double r144819 = r144816 * r144818;
        double r144820 = sqrt(r144819);
        double r144821 = 8.420291904348501e-153;
        bool r144822 = r144791 <= r144821;
        double r144823 = r144802 * r144803;
        double r144824 = pow(r144798, r144823);
        double r144825 = r144791 * r144824;
        double r144826 = r144812 * r144825;
        double r144827 = fma(r144795, r144799, r144826);
        double r144828 = r144794 - r144827;
        double r144829 = r144828 * r144817;
        double r144830 = r144829 * r144810;
        double r144831 = sqrt(r144830);
        double r144832 = r144791 * r144809;
        double r144833 = cbrt(r144813);
        double r144834 = r144833 * r144833;
        double r144835 = r144834 * r144833;
        double r144836 = r144832 * r144835;
        double r144837 = fma(r144795, r144799, r144836);
        double r144838 = r144794 - r144837;
        double r144839 = r144838 * r144818;
        double r144840 = sqrt(r144839);
        double r144841 = r144822 ? r144831 : r144840;
        double r144842 = r144793 ? r144820 : r144841;
        return r144842;
}

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 3 regimes

2. if n < -2.0370585079861942e-134

   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 33.2

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

   4. Applied *-un-lft-identity 33.2

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

   5. Applied times-frac 30.8

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

   6. Simplified 30.8

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

   8. Applied sqr-pow 30.8

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

   9. Applied associate-*r* 29.9

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

   11. Applied associate-*l* 29.0

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

   13. Applied *-un-lft-identity 29.0

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

   14. Applied add-cube-cbrt 29.0

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

   15. Applied times-frac 29.0

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

   16. Applied unpow-prod-down 29.0

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

   17. Applied associate-*r* 29.6

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

   18. Simplified 29.6

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

   if -2.0370585079861942e-134 < n < 8.420291904348501e-153

   1. Initial program 38.6

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

   2. Simplified 38.6

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

   4. Applied *-un-lft-identity 38.6

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

   5. Applied times-frac 35.7

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

   6. Simplified 35.7

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

   8. Applied sqr-pow 35.7

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

   9. Applied associate-*r* 34.8

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

   11. Applied associate-*l* 36.2

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

   13. Applied associate-*r* 31.2

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

   14. Simplified 31.4

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

   if 8.420291904348501e-153 < n

   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. Simplified 32.7

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

   4. Applied *-un-lft-identity 32.7

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

   5. Applied times-frac 30.2

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

   6. Simplified 30.2

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

   8. Applied sqr-pow 30.2

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

   9. Applied associate-*r* 29.5

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

   11. Applied associate-*l* 28.8

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

   13. Applied add-cube-cbrt 28.8

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

4. Final simplification 30.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.037058507986194199807262357510988881335 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\left(n \cdot {\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 8.420291904348501272367972443482728416614 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)} \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)}\right) \cdot \sqrt[3]{{\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))