Toniolo and Linder, Equation (7)

Average Error: 42.9 → 10.3

Time: 10.3s

Precision: 64

Math

\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6.1135263195832389 \cdot 10^{48}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -7.40273667700519053 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\left({\left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}}{\sqrt[3]{x}}\right) \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le -6.9713348016436594 \cdot 10^{-265}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 5.62316094137459942 \cdot 10^{76}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\left({\left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}}{\sqrt[3]{x}}\right) \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]

C

double f(double x, double l, double t) {
        double r40795 = 2.0;
        double r40796 = sqrt(r40795);
        double r40797 = t;
        double r40798 = r40796 * r40797;
        double r40799 = x;
        double r40800 = 1.0;
        double r40801 = r40799 + r40800;
        double r40802 = r40799 - r40800;
        double r40803 = r40801 / r40802;
        double r40804 = l;
        double r40805 = r40804 * r40804;
        double r40806 = r40797 * r40797;
        double r40807 = r40795 * r40806;
        double r40808 = r40805 + r40807;
        double r40809 = r40803 * r40808;
        double r40810 = r40809 - r40805;
        double r40811 = sqrt(r40810);
        double r40812 = r40798 / r40811;
        return r40812;
}

↓

double f(double x, double l, double t) {
        double r40813 = t;
        double r40814 = -6.113526319583239e+48;
        bool r40815 = r40813 <= r40814;
        double r40816 = 2.0;
        double r40817 = sqrt(r40816);
        double r40818 = r40817 * r40813;
        double r40819 = 3.0;
        double r40820 = pow(r40817, r40819);
        double r40821 = x;
        double r40822 = 2.0;
        double r40823 = pow(r40821, r40822);
        double r40824 = r40820 * r40823;
        double r40825 = r40813 / r40824;
        double r40826 = r40817 * r40823;
        double r40827 = r40813 / r40826;
        double r40828 = r40825 - r40827;
        double r40829 = r40816 * r40828;
        double r40830 = r40829 - r40818;
        double r40831 = r40817 * r40821;
        double r40832 = r40813 / r40831;
        double r40833 = r40816 * r40832;
        double r40834 = r40830 - r40833;
        double r40835 = r40818 / r40834;
        double r40836 = -7.40273667700519e-119;
        bool r40837 = r40813 <= r40836;
        double r40838 = 4.0;
        double r40839 = pow(r40813, r40822);
        double r40840 = r40839 / r40821;
        double r40841 = r40838 * r40840;
        double r40842 = l;
        double r40843 = cbrt(r40842);
        double r40844 = r40843 * r40843;
        double r40845 = cbrt(r40844);
        double r40846 = 4.0;
        double r40847 = pow(r40845, r40846);
        double r40848 = cbrt(r40843);
        double r40849 = pow(r40848, r40846);
        double r40850 = cbrt(r40821);
        double r40851 = r40849 / r40850;
        double r40852 = r40847 * r40851;
        double r40853 = fabs(r40843);
        double r40854 = r40853 / r40850;
        double r40855 = r40852 * r40854;
        double r40856 = pow(r40843, r40822);
        double r40857 = sqrt(r40856);
        double r40858 = r40857 / r40850;
        double r40859 = r40855 * r40858;
        double r40860 = r40839 + r40859;
        double r40861 = r40816 * r40860;
        double r40862 = r40841 + r40861;
        double r40863 = sqrt(r40862);
        double r40864 = r40818 / r40863;
        double r40865 = -6.971334801643659e-265;
        bool r40866 = r40813 <= r40865;
        double r40867 = 5.6231609413745994e+76;
        bool r40868 = r40813 <= r40867;
        double r40869 = r40813 * r40817;
        double r40870 = r40833 + r40869;
        double r40871 = r40816 * r40825;
        double r40872 = r40870 - r40871;
        double r40873 = r40818 / r40872;
        double r40874 = r40868 ? r40864 : r40873;
        double r40875 = r40866 ? r40835 : r40874;
        double r40876 = r40837 ? r40864 : r40875;
        double r40877 = r40815 ? r40835 : r40876;
        return r40877;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 3 regimes

2. if t < -6.113526319583239e+48 or -7.40273667700519e-119 < t < -6.971334801643659e-265

   1. Initial program 46.3

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around -inf 11.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)\right)}}\]

   3. Simplified 11.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

   if -6.113526319583239e+48 < t < -7.40273667700519e-119 or -6.971334801643659e-265 < t < 5.6231609413745994e+76

   1. Initial program 37.5

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around inf 16.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

   3. Simplified 16.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{x}\right)}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 16.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{\color{blue}{1 \cdot x}}\right)}}\]

   6. Applied add-cube-cbrt 16.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\color{blue}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}\right)}}^{2}}{1 \cdot x}\right)}}\]

   7. Applied unpow-prod-down 16.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\color{blue}{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}}{1 \cdot x}\right)}}\]

   8. Applied times-frac 13.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\frac{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}}\right)}}\]

   9. Simplified 13.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}\right)}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 13.5

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + {\left(\sqrt[3]{\ell}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right)}}\]

   12. Applied add-sqr-sqrt 13.5

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + {\left(\sqrt[3]{\ell}\right)}^{4} \cdot \frac{\color{blue}{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)}}\]

   13. Applied times-frac 13.5

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + {\left(\sqrt[3]{\ell}\right)}^{4} \cdot \color{blue}{\left(\frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}\right)}}\]

   14. Applied associate-*r* 13.5

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\left({\left(\sqrt[3]{\ell}\right)}^{4} \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}}\right)}}\]

   15. Simplified 13.5

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{4}}{\sqrt[3]{x}} \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right)} \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\]
   16. Using strategy rm

   17. Applied *-un-lft-identity 13.5

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\frac{{\left(\sqrt[3]{\ell}\right)}^{4}}{\sqrt[3]{\color{blue}{1 \cdot x}}} \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\]

   18. Applied cbrt-prod 13.5

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\frac{{\left(\sqrt[3]{\ell}\right)}^{4}}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{x}}} \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\]

   19. Applied add-cube-cbrt 13.5

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\frac{{\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right)}^{4}}{\sqrt[3]{1} \cdot \sqrt[3]{x}} \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\]

   20. Applied cbrt-prod 13.6

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\frac{{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}\right)}}^{4}}{\sqrt[3]{1} \cdot \sqrt[3]{x}} \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\]

   21. Applied unpow-prod-down 13.6

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\frac{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{4} \cdot {\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}}}{\sqrt[3]{1} \cdot \sqrt[3]{x}} \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\]

   22. Applied times-frac 12.6

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\color{blue}{\left(\frac{{\left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{4}}{\sqrt[3]{1}} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}}{\sqrt[3]{x}}\right)} \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\]

   23. Simplified 12.6

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}}{\sqrt[3]{x}}\right) \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\]

   if 5.6231609413745994e+76 < t

   1. Initial program 48.2

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

   2. Taylor expanded around inf 47.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

   3. Simplified 47.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{x}\right)}}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 47.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{\color{blue}{1 \cdot x}}\right)}}\]

   6. Applied add-cube-cbrt 47.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\color{blue}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}\right)}}^{2}}{1 \cdot x}\right)}}\]

   7. Applied unpow-prod-down 47.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\color{blue}{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}}{1 \cdot x}\right)}}\]

   8. Applied times-frac 45.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\frac{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}}\right)}}\]

   9. Simplified 45.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{{\left(\sqrt[3]{\ell}\right)}^{4}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{x}\right)}}\]

   10. Taylor expanded around inf 3.3

       \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.1135263195832389 \cdot 10^{48}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -7.40273667700519053 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\left({\left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}}{\sqrt[3]{x}}\right) \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le -6.9713348016436594 \cdot 10^{-265}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 5.62316094137459942 \cdot 10^{76}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\left({\left(\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}}{\sqrt[3]{x}}\right) \cdot \frac{\left|\sqrt[3]{\ell}\right|}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))