Toniolo and Linder, Equation (7)

Average Error: 43.4 → 9.9

Time: 29.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.488419163648193597366161151316767145753 \cdot 10^{95}:\\ \;\;\;\;\frac{\sqrt{2}}{-\mathsf{fma}\left(\frac{2}{\sqrt{2}}, \frac{t}{x}, \sqrt{2} \cdot t\right)} \cdot t\\ \mathbf{elif}\;t \le 3.654137014915323109348805874635869440815 \cdot 10^{-285}:\\ \;\;\;\;t \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, \frac{\left(t \cdot t\right) \cdot 4}{x}\right)\right)}}\right)\\ \mathbf{elif}\;t \le 2.084962264332403679721070208359607902211 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\sqrt{2}}}{\mathsf{fma}\left(\frac{\frac{t}{\sqrt{2}}}{x}, 2, \sqrt{2} \cdot t\right)} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot t\\ \mathbf{elif}\;t \le 1.814580254256021431914078254441608287695 \cdot 10^{149}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, \frac{\left(t \cdot t\right) \cdot 4}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(2, \frac{\frac{t}{\sqrt{2}}}{x} - \frac{\frac{t}{x \cdot x}}{2 \cdot \sqrt{2}}, \sqrt{2} \cdot t\right)}\\ \end{array}\]

C

double f(double x, double l, double t) {
        double r42782 = 2.0;
        double r42783 = sqrt(r42782);
        double r42784 = t;
        double r42785 = r42783 * r42784;
        double r42786 = x;
        double r42787 = 1.0;
        double r42788 = r42786 + r42787;
        double r42789 = r42786 - r42787;
        double r42790 = r42788 / r42789;
        double r42791 = l;
        double r42792 = r42791 * r42791;
        double r42793 = r42784 * r42784;
        double r42794 = r42782 * r42793;
        double r42795 = r42792 + r42794;
        double r42796 = r42790 * r42795;
        double r42797 = r42796 - r42792;
        double r42798 = sqrt(r42797);
        double r42799 = r42785 / r42798;
        return r42799;
}

↓

double f(double x, double l, double t) {
        double r42800 = t;
        double r42801 = -5.488419163648194e+95;
        bool r42802 = r42800 <= r42801;
        double r42803 = 2.0;
        double r42804 = sqrt(r42803);
        double r42805 = r42803 / r42804;
        double r42806 = x;
        double r42807 = r42800 / r42806;
        double r42808 = r42804 * r42800;
        double r42809 = fma(r42805, r42807, r42808);
        double r42810 = -r42809;
        double r42811 = r42804 / r42810;
        double r42812 = r42811 * r42800;
        double r42813 = 3.654137014915323e-285;
        bool r42814 = r42800 <= r42813;
        double r42815 = cbrt(r42804);
        double r42816 = r42815 * r42815;
        double r42817 = r42800 * r42800;
        double r42818 = l;
        double r42819 = r42818 / r42806;
        double r42820 = r42818 * r42819;
        double r42821 = 4.0;
        double r42822 = r42817 * r42821;
        double r42823 = r42822 / r42806;
        double r42824 = fma(r42803, r42820, r42823);
        double r42825 = fma(r42803, r42817, r42824);
        double r42826 = sqrt(r42825);
        double r42827 = r42815 / r42826;
        double r42828 = r42816 * r42827;
        double r42829 = r42800 * r42828;
        double r42830 = 2.0849622643324037e-159;
        bool r42831 = r42800 <= r42830;
        double r42832 = r42800 / r42804;
        double r42833 = r42832 / r42806;
        double r42834 = fma(r42833, r42803, r42808);
        double r42835 = r42815 / r42834;
        double r42836 = r42835 * r42816;
        double r42837 = r42836 * r42800;
        double r42838 = 1.8145802542560214e+149;
        bool r42839 = r42800 <= r42838;
        double r42840 = r42803 / r42825;
        double r42841 = sqrt(r42840);
        double r42842 = r42800 * r42841;
        double r42843 = r42806 * r42806;
        double r42844 = r42800 / r42843;
        double r42845 = r42803 * r42804;
        double r42846 = r42844 / r42845;
        double r42847 = r42833 - r42846;
        double r42848 = fma(r42803, r42847, r42808);
        double r42849 = r42804 / r42848;
        double r42850 = r42800 * r42849;
        double r42851 = r42839 ? r42842 : r42850;
        double r42852 = r42831 ? r42837 : r42851;
        double r42853 = r42814 ? r42829 : r42852;
        double r42854 = r42802 ? r42812 : r42853;
        return r42854;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 5 regimes

2. if t < -5.488419163648194e+95

   1. Initial program 50.7

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

   2. Simplified 50.7

      \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\ell, \ell, \left(t \cdot t\right) \cdot 2\right) \cdot \frac{1 + x}{x - 1} - \ell \cdot \ell}}}\]

   3. Taylor expanded around inf 50.5

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

   4. Simplified 50.5

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

   5. Taylor expanded around 0 50.5

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

   6. Simplified 49.0

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

   7. Taylor expanded around -inf 3.6

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

   8. Simplified 3.6

      \[\leadsto t \cdot \frac{\sqrt{2}}{\color{blue}{-\mathsf{fma}\left(\frac{2}{\sqrt{2}}, \frac{t}{x}, \sqrt{2} \cdot t\right)}}\]

   if -5.488419163648194e+95 < t < 3.654137014915323e-285

   1. Initial program 40.0

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

   2. Simplified 39.9

      \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\ell, \ell, \left(t \cdot t\right) \cdot 2\right) \cdot \frac{1 + x}{x - 1} - \ell \cdot \ell}}}\]

   3. Taylor expanded around inf 18.2

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

   4. Simplified 18.2

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

   5. Taylor expanded around 0 18.2

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

   6. Simplified 14.4

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

   8. Applied *-un-lft-identity 14.4

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

   9. Applied sqrt-prod 14.4

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

   10. Applied add-cube-cbrt 14.4

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

   11. Applied times-frac 14.4

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

   12. Simplified 14.4

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

   13. Simplified 14.4

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

   if 3.654137014915323e-285 < t < 2.0849622643324037e-159

   1. Initial program 62.8

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

   2. Simplified 62.8

      \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\ell, \ell, \left(t \cdot t\right) \cdot 2\right) \cdot \frac{1 + x}{x - 1} - \ell \cdot \ell}}}\]

   3. Taylor expanded around inf 33.7

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

   4. Simplified 33.7

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

   5. Taylor expanded around 0 33.7

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

   6. Simplified 31.2

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

   8. Applied *-un-lft-identity 31.2

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

   9. Applied sqrt-prod 31.2

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

   10. Applied add-cube-cbrt 31.2

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

   11. Applied times-frac 31.2

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

   12. Simplified 31.2

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

   13. Simplified 31.2

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

   14. Taylor expanded around inf 37.0

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

   15. Simplified 37.0

       \[\leadsto t \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{\color{blue}{\mathsf{fma}\left(\frac{\frac{t}{\sqrt{2}}}{x}, 2, t \cdot \sqrt{2}\right)}}\right)\]

   if 2.0849622643324037e-159 < t < 1.8145802542560214e+149

   1. Initial program 23.8

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

   2. Simplified 23.8

      \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\ell, \ell, \left(t \cdot t\right) \cdot 2\right) \cdot \frac{1 + x}{x - 1} - \ell \cdot \ell}}}\]

   3. Taylor expanded around inf 10.3

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

   4. Simplified 10.3

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

   5. Taylor expanded around 0 10.3

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

   6. Simplified 5.1

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

   8. Applied sqrt-undiv 5.5

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

   9. Simplified 5.5

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

   if 1.8145802542560214e+149 < t

   1. Initial program 61.0

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

   2. Simplified 61.0

      \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\ell, \ell, \left(t \cdot t\right) \cdot 2\right) \cdot \frac{1 + x}{x - 1} - \ell \cdot \ell}}}\]

   3. Taylor expanded around inf 61.5

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

   4. Simplified 61.5

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

   5. Taylor expanded around inf 2.1

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

   6. Simplified 2.1

      \[\leadsto t \cdot \frac{\sqrt{2}}{\color{blue}{\mathsf{fma}\left(2, \frac{\frac{t}{\sqrt{2}}}{x} - \frac{\frac{t}{x \cdot x}}{\sqrt{2} \cdot 2}, t \cdot \sqrt{2}\right)}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.488419163648193597366161151316767145753 \cdot 10^{95}:\\ \;\;\;\;\frac{\sqrt{2}}{-\mathsf{fma}\left(\frac{2}{\sqrt{2}}, \frac{t}{x}, \sqrt{2} \cdot t\right)} \cdot t\\ \mathbf{elif}\;t \le 3.654137014915323109348805874635869440815 \cdot 10^{-285}:\\ \;\;\;\;t \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{\sqrt{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, \frac{\left(t \cdot t\right) \cdot 4}{x}\right)\right)}}\right)\\ \mathbf{elif}\;t \le 2.084962264332403679721070208359607902211 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\sqrt{2}}}{\mathsf{fma}\left(\frac{\frac{t}{\sqrt{2}}}{x}, 2, \sqrt{2} \cdot t\right)} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\right) \cdot t\\ \mathbf{elif}\;t \le 1.814580254256021431914078254441608287695 \cdot 10^{149}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, t \cdot t, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, \frac{\left(t \cdot t\right) \cdot 4}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(2, \frac{\frac{t}{\sqrt{2}}}{x} - \frac{\frac{t}{x \cdot x}}{2 \cdot \sqrt{2}}, \sqrt{2} \cdot t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))