Toniolo and Linder, Equation (13)

Average Error: 33.7 → 23.9

Time: 48.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 1.0022495870274159 \cdot 10^{-154}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot 2} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}}\right)}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right)}^{\frac{3}{2}}\\ \mathbf{elif}\;t \le 2.6158530040231545 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot 2} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}}\right)}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right)}^{\frac{3}{2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2490815 = 2.0;
        double r2490816 = n;
        double r2490817 = r2490815 * r2490816;
        double r2490818 = U;
        double r2490819 = r2490817 * r2490818;
        double r2490820 = t;
        double r2490821 = l;
        double r2490822 = r2490821 * r2490821;
        double r2490823 = Om;
        double r2490824 = r2490822 / r2490823;
        double r2490825 = r2490815 * r2490824;
        double r2490826 = r2490820 - r2490825;
        double r2490827 = r2490821 / r2490823;
        double r2490828 = pow(r2490827, r2490815);
        double r2490829 = r2490816 * r2490828;
        double r2490830 = U_;
        double r2490831 = r2490818 - r2490830;
        double r2490832 = r2490829 * r2490831;
        double r2490833 = r2490826 - r2490832;
        double r2490834 = r2490819 * r2490833;
        double r2490835 = sqrt(r2490834);
        return r2490835;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2490836 = t;
        double r2490837 = 1.0022495870274159e-154;
        bool r2490838 = r2490836 <= r2490837;
        double r2490839 = U;
        double r2490840 = 2.0;
        double r2490841 = r2490839 * r2490840;
        double r2490842 = cbrt(r2490841);
        double r2490843 = U_;
        double r2490844 = r2490843 - r2490839;
        double r2490845 = l;
        double r2490846 = cbrt(r2490845);
        double r2490847 = n;
        double r2490848 = cbrt(r2490847);
        double r2490849 = r2490846 * r2490848;
        double r2490850 = Om;
        double r2490851 = cbrt(r2490850);
        double r2490852 = r2490849 / r2490851;
        double r2490853 = r2490852 * r2490852;
        double r2490854 = r2490852 * r2490853;
        double r2490855 = r2490850 / r2490845;
        double r2490856 = r2490854 / r2490855;
        double r2490857 = r2490845 / r2490855;
        double r2490858 = -2.0;
        double r2490859 = fma(r2490857, r2490858, r2490836);
        double r2490860 = fma(r2490844, r2490856, r2490859);
        double r2490861 = r2490860 * r2490847;
        double r2490862 = cbrt(r2490861);
        double r2490863 = r2490842 * r2490862;
        double r2490864 = 1.5;
        double r2490865 = pow(r2490863, r2490864);
        double r2490866 = 2.6158530040231545e-73;
        bool r2490867 = r2490836 <= r2490866;
        double r2490868 = r2490847 / r2490855;
        double r2490869 = r2490868 / r2490855;
        double r2490870 = fma(r2490844, r2490869, r2490859);
        double r2490871 = r2490847 * r2490841;
        double r2490872 = r2490870 * r2490871;
        double r2490873 = sqrt(r2490872);
        double r2490874 = r2490867 ? r2490873 : r2490865;
        double r2490875 = r2490838 ? r2490865 : r2490874;
        return r2490875;
}

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 2 regimes

2. if t < 1.0022495870274159e-154 or 2.6158530040231545e-73 < t

   1. Initial program 33.6

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

   2. Simplified 29.8

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

   4. Applied add-cube-cbrt 29.9

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

   5. Applied add-cube-cbrt 29.9

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

   6. Applied times-frac 29.9

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

   7. Applied add-cube-cbrt 29.9

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

   8. Applied times-frac 29.9

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

   9. Simplified 29.9

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

   10. Simplified 29.9

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

   12. Applied add-cube-cbrt 30.2

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

   14. Applied pow1 30.2

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

   15. Applied pow1 30.2

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

   16. Applied pow1 30.2

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

   17. Applied pow-prod-up 30.2

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

   18. Applied pow-prod-up 30.2

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

   19. Applied sqrt-pow1 30.2

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

   20. Simplified 30.2

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

   22. Applied cbrt-prod 23.5

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

   if 1.0022495870274159e-154 < t < 2.6158530040231545e-73

   1. Initial program 34.4

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

   2. Simplified 31.0

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

   4. Applied associate-*r* 30.4

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

4. Final simplification 23.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.0022495870274159 \cdot 10^{-154}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot 2} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}}\right)}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right)}^{\frac{3}{2}}\\ \mathbf{elif}\;t \le 2.6158530040231545 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot 2} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}}\right)}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right)}^{\frac{3}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(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*))))))