Toniolo and Linder, Equation (13)

Average Error: 34.9 → 29.8

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 -1.77082420255664932142459979130796319176 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(\left(U \cdot \sqrt[3]{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\right)}^{3}}\right) \cdot \sqrt[3]{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\right)\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \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)}}\\ \mathbf{elif}\;t \le 1.038166891895459751679228703524086803936 \cdot 10^{145}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \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)}\\ \mathbf{elif}\;t \le 1.847000503161857541382626154334586216151 \cdot 10^{147}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \left(\sqrt[3]{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right) \cdot \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, \left(U - U*\right) \cdot n, t\right)\right)} \cdot \left(\left(U \cdot \sqrt[3]{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n, U - U*, t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right)}\right)\right)\right)}}{\sqrt{\sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n, U - U*, t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, \left(U - U*\right) \cdot n, t\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r158721 = 2.0;
        double r158722 = n;
        double r158723 = r158721 * r158722;
        double r158724 = U;
        double r158725 = r158723 * r158724;
        double r158726 = t;
        double r158727 = l;
        double r158728 = r158727 * r158727;
        double r158729 = Om;
        double r158730 = r158728 / r158729;
        double r158731 = r158721 * r158730;
        double r158732 = r158726 - r158731;
        double r158733 = r158727 / r158729;
        double r158734 = pow(r158733, r158721);
        double r158735 = r158722 * r158734;
        double r158736 = U_;
        double r158737 = r158724 - r158736;
        double r158738 = r158735 * r158737;
        double r158739 = r158732 - r158738;
        double r158740 = r158725 * r158739;
        double r158741 = sqrt(r158740);
        return r158741;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r158742 = t;
        double r158743 = -1.7708242025566493e-147;
        bool r158744 = r158742 <= r158743;
        double r158745 = 2.0;
        double r158746 = n;
        double r158747 = r158745 * r158746;
        double r158748 = U;
        double r158749 = l;
        double r158750 = Om;
        double r158751 = r158750 / r158749;
        double r158752 = r158749 / r158751;
        double r158753 = U_;
        double r158754 = r158748 - r158753;
        double r158755 = r158749 / r158750;
        double r158756 = 2.0;
        double r158757 = r158745 / r158756;
        double r158758 = r158756 * r158757;
        double r158759 = pow(r158755, r158758);
        double r158760 = r158746 * r158759;
        double r158761 = r158754 * r158760;
        double r158762 = fma(r158745, r158752, r158761);
        double r158763 = r158742 - r158762;
        double r158764 = cbrt(r158763);
        double r158765 = 3.0;
        double r158766 = pow(r158764, r158765);
        double r158767 = cbrt(r158766);
        double r158768 = r158748 * r158767;
        double r158769 = r158768 * r158764;
        double r158770 = r158747 * r158769;
        double r158771 = r158745 * r158752;
        double r158772 = r158742 - r158771;
        double r158773 = pow(r158755, r158757);
        double r158774 = r158746 * r158773;
        double r158775 = r158774 * r158773;
        double r158776 = r158775 * r158754;
        double r158777 = r158772 - r158776;
        double r158778 = cbrt(r158777);
        double r158779 = r158770 * r158778;
        double r158780 = sqrt(r158779);
        double r158781 = 1.0381668918954598e+145;
        bool r158782 = r158742 <= r158781;
        double r158783 = r158747 * r158748;
        double r158784 = r158783 * r158777;
        double r158785 = sqrt(r158784);
        double r158786 = 1.8470005031618575e+147;
        bool r158787 = r158742 <= r158786;
        double r158788 = r158754 * r158746;
        double r158789 = fma(r158759, r158788, r158742);
        double r158790 = fma(r158745, r158752, r158789);
        double r158791 = r158763 * r158790;
        double r158792 = cbrt(r158791);
        double r158793 = r158748 * r158764;
        double r158794 = r158759 * r158746;
        double r158795 = fma(r158794, r158754, r158772);
        double r158796 = r158795 * r158763;
        double r158797 = cbrt(r158796);
        double r158798 = r158793 * r158797;
        double r158799 = r158792 * r158798;
        double r158800 = r158745 * r158799;
        double r158801 = r158746 * r158800;
        double r158802 = sqrt(r158801);
        double r158803 = cbrt(r158795);
        double r158804 = cbrt(r158790);
        double r158805 = r158803 * r158804;
        double r158806 = sqrt(r158805);
        double r158807 = r158802 / r158806;
        double r158808 = sqrt(r158783);
        double r158809 = pow(r158755, r158745);
        double r158810 = r158754 * r158809;
        double r158811 = r158746 * r158810;
        double r158812 = fma(r158745, r158752, r158811);
        double r158813 = r158742 - r158812;
        double r158814 = sqrt(r158813);
        double r158815 = r158808 * r158814;
        double r158816 = r158787 ? r158807 : r158815;
        double r158817 = r158782 ? r158785 : r158816;
        double r158818 = r158744 ? r158780 : r158817;
        return r158818;
}

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 < -1.7708242025566493e-147

   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. Using strategy rm

   3. Applied associate-/l* 30.6

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

   5. Applied sqr-pow 30.6

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

   6. Applied associate-*r* 30.0

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

   8. Applied add-cube-cbrt 30.3

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

   9. Applied associate-*r* 30.3

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

   10. Simplified 29.8

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

   12. Applied add-cbrt-cube 29.8

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

   13. Simplified 29.8

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

   if -1.7708242025566493e-147 < t < 1.0381668918954598e+145

   1. Initial program 34.7

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

   3. Applied associate-/l* 31.9

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

   5. Applied sqr-pow 31.9

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

   6. Applied associate-*r* 30.9

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

   if 1.0381668918954598e+145 < t < 1.8470005031618575e+147

   1. Initial program 42.3

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

   3. Applied associate-/l* 38.7

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

   5. Applied sqr-pow 38.7

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

   6. Applied associate-*r* 38.7

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

   8. Applied add-cube-cbrt 38.9

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

   9. Applied associate-*r* 38.9

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

   10. Simplified 38.0

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

   12. Applied flip-- 38.0

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

   13. Applied cbrt-div 37.8

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

   14. Applied flip-- 37.8

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

   15. Applied cbrt-div 37.9

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

   16. Applied associate-*r/ 37.8

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

   17. Applied associate-*r/ 37.9

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

   18. Applied frac-times 33.4

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

   19. Applied sqrt-div 33.4

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

   20. Simplified 42.8

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

   21. Simplified 42.8

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

   if 1.8470005031618575e+147 < t

   1. Initial program 38.2

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   2. 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 sqrt-prod 25.1

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

   6. Simplified 26.0

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

4. Final simplification 29.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.77082420255664932142459979130796319176 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(\left(U \cdot \sqrt[3]{{\left(\sqrt[3]{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\right)}^{3}}\right) \cdot \sqrt[3]{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\right)\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \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)}}\\ \mathbf{elif}\;t \le 1.038166891895459751679228703524086803936 \cdot 10^{145}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \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)}\\ \mathbf{elif}\;t \le 1.847000503161857541382626154334586216151 \cdot 10^{147}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \left(\sqrt[3]{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right) \cdot \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, \left(U - U*\right) \cdot n, t\right)\right)} \cdot \left(\left(U \cdot \sqrt[3]{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n, U - U*, t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right)}\right)\right)\right)}}{\sqrt{\sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n, U - U*, t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, \left(U - U*\right) \cdot n, t\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, n \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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*))))))