Toniolo and Linder, Equation (13)

Average Error: 34.6 → 27.0

Time: 2.5m

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 -7.411448247083684376522735829909480721372 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}\\ \mathbf{elif}\;U \le 2.469810196486197144576887320962391000635 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r478771 = 2.0;
        double r478772 = n;
        double r478773 = r478771 * r478772;
        double r478774 = U;
        double r478775 = r478773 * r478774;
        double r478776 = t;
        double r478777 = l;
        double r478778 = r478777 * r478777;
        double r478779 = Om;
        double r478780 = r478778 / r478779;
        double r478781 = r478771 * r478780;
        double r478782 = r478776 - r478781;
        double r478783 = r478777 / r478779;
        double r478784 = pow(r478783, r478771);
        double r478785 = r478772 * r478784;
        double r478786 = U_;
        double r478787 = r478774 - r478786;
        double r478788 = r478785 * r478787;
        double r478789 = r478782 - r478788;
        double r478790 = r478775 * r478789;
        double r478791 = sqrt(r478790);
        return r478791;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r478792 = U;
        double r478793 = -7.411448247083684e-46;
        bool r478794 = r478792 <= r478793;
        double r478795 = 2.0;
        double r478796 = n;
        double r478797 = r478795 * r478796;
        double r478798 = l;
        double r478799 = Om;
        double r478800 = r478798 / r478799;
        double r478801 = 2.0;
        double r478802 = r478795 / r478801;
        double r478803 = pow(r478800, r478802);
        double r478804 = r478796 * r478803;
        double r478805 = r478804 * r478803;
        double r478806 = U_;
        double r478807 = r478806 - r478792;
        double r478808 = r478805 * r478807;
        double r478809 = r478795 * r478798;
        double r478810 = r478800 * r478809;
        double r478811 = r478808 - r478810;
        double r478812 = t;
        double r478813 = r478811 + r478812;
        double r478814 = r478797 * r478813;
        double r478815 = r478792 * r478814;
        double r478816 = sqrt(r478815);
        double r478817 = 2.469810196486197e-303;
        bool r478818 = r478792 <= r478817;
        double r478819 = cbrt(r478799);
        double r478820 = r478798 / r478819;
        double r478821 = pow(r478820, r478802);
        double r478822 = r478821 * r478803;
        double r478823 = 1.0;
        double r478824 = r478819 * r478819;
        double r478825 = r478823 / r478824;
        double r478826 = pow(r478825, r478802);
        double r478827 = r478796 * r478826;
        double r478828 = r478822 * r478827;
        double r478829 = r478828 * r478807;
        double r478830 = r478810 - r478812;
        double r478831 = r478829 - r478830;
        double r478832 = r478792 * r478831;
        double r478833 = r478797 * r478832;
        double r478834 = cbrt(r478833);
        double r478835 = r478834 * r478834;
        double r478836 = r478835 * r478834;
        double r478837 = sqrt(r478836);
        double r478838 = r478831 * r478797;
        double r478839 = sqrt(r478838);
        double r478840 = sqrt(r478792);
        double r478841 = r478839 * r478840;
        double r478842 = r478818 ? r478837 : r478841;
        double r478843 = r478794 ? r478816 : r478842;
        return r478843;
}

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 U < -7.411448247083684e-46

   1. Initial program 29.8

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

   2. Simplified 28.4

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

   4. Applied sqr-pow 28.4

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

   5. Applied associate-*r* 27.3

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

   if -7.411448247083684e-46 < U < 2.469810196486197e-303

   1. Initial program 39.3

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

   2. Simplified 35.6

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

   4. Applied sqr-pow 35.6

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

   5. Applied associate-*r* 34.6

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

   7. Applied add-cube-cbrt 34.7

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

   8. Applied *-un-lft-identity 34.7

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

   9. Applied times-frac 34.7

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

   10. Applied unpow-prod-down 34.7

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

   11. Applied associate-*r* 34.9

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

   12. Simplified 34.9

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

   14. Applied add-cube-cbrt 35.1

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

   15. Simplified 37.2

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

   16. Simplified 32.9

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

   if 2.469810196486197e-303 < U

   1. Initial program 34.2

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

   2. Simplified 31.3

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

   4. Applied sqr-pow 31.3

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

   5. Applied associate-*r* 30.2

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

   7. Applied add-cube-cbrt 30.3

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

   8. Applied *-un-lft-identity 30.3

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

   9. Applied times-frac 30.3

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

   10. Applied unpow-prod-down 30.3

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

   11. Applied associate-*r* 30.3

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

   12. Simplified 30.3

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

   14. Applied sqrt-prod 23.2

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

   15. Simplified 23.7

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

4. Final simplification 27.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -7.411448247083684376522735829909480721372 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\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) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}\\ \mathbf{elif}\;U \le 2.469810196486197144576887320962391000635 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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*))))))