Toniolo and Linder, Equation (10+)

Average Error: 31.4 → 7.2

Time: 5.9m

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.079547548008425 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{\tan k}}}{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}}\\ \mathbf{elif}\;t \le 8.337567466703915 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\left(\frac{\sin k}{\frac{\ell}{k \cdot k}} + \frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot 2\right) \cdot \frac{1}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt[3]{\tan k}}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}} \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}}\right)} \cdot \left(\frac{\sqrt[3]{\cos k}}{\frac{1}{\sqrt[3]{\frac{\ell}{t}}}} \cdot \frac{\frac{1}{\sqrt[3]{\tan k}}}{\left(\frac{1}{\sqrt[3]{\frac{\ell}{t}}} \cdot \frac{1}{\sqrt[3]{\frac{\ell}{t}}}\right) \cdot \sqrt[3]{\sin k}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r8920811 = 2.0;
        double r8920812 = t;
        double r8920813 = 3.0;
        double r8920814 = pow(r8920812, r8920813);
        double r8920815 = l;
        double r8920816 = r8920815 * r8920815;
        double r8920817 = r8920814 / r8920816;
        double r8920818 = k;
        double r8920819 = sin(r8920818);
        double r8920820 = r8920817 * r8920819;
        double r8920821 = tan(r8920818);
        double r8920822 = r8920820 * r8920821;
        double r8920823 = 1.0;
        double r8920824 = r8920818 / r8920812;
        double r8920825 = pow(r8920824, r8920811);
        double r8920826 = r8920823 + r8920825;
        double r8920827 = r8920826 + r8920823;
        double r8920828 = r8920822 * r8920827;
        double r8920829 = r8920811 / r8920828;
        return r8920829;
}

↓

double f(double t, double l, double k) {
        double r8920830 = t;
        double r8920831 = -1.079547548008425e-56;
        bool r8920832 = r8920830 <= r8920831;
        double r8920833 = 2.0;
        double r8920834 = sqrt(r8920833);
        double r8920835 = k;
        double r8920836 = tan(r8920835);
        double r8920837 = cbrt(r8920836);
        double r8920838 = r8920837 * r8920837;
        double r8920839 = r8920834 / r8920838;
        double r8920840 = 1.0;
        double r8920841 = l;
        double r8920842 = r8920841 / r8920830;
        double r8920843 = r8920840 / r8920842;
        double r8920844 = r8920839 / r8920843;
        double r8920845 = r8920834 / r8920837;
        double r8920846 = r8920835 / r8920830;
        double r8920847 = fma(r8920846, r8920846, r8920833);
        double r8920848 = sin(r8920835);
        double r8920849 = r8920842 / r8920848;
        double r8920850 = r8920830 / r8920849;
        double r8920851 = r8920847 * r8920850;
        double r8920852 = r8920845 / r8920851;
        double r8920853 = r8920844 * r8920852;
        double r8920854 = 8.337567466703915e+16;
        bool r8920855 = r8920830 <= r8920854;
        double r8920856 = r8920833 / r8920836;
        double r8920857 = r8920835 * r8920835;
        double r8920858 = r8920841 / r8920857;
        double r8920859 = r8920848 / r8920858;
        double r8920860 = r8920830 * r8920830;
        double r8920861 = r8920841 / r8920848;
        double r8920862 = r8920860 / r8920861;
        double r8920863 = r8920862 * r8920833;
        double r8920864 = r8920859 + r8920863;
        double r8920865 = r8920864 * r8920843;
        double r8920866 = r8920856 / r8920865;
        double r8920867 = r8920833 / r8920837;
        double r8920868 = cbrt(r8920851);
        double r8920869 = r8920868 * r8920868;
        double r8920870 = r8920868 * r8920869;
        double r8920871 = r8920867 / r8920870;
        double r8920872 = cos(r8920835);
        double r8920873 = cbrt(r8920872);
        double r8920874 = cbrt(r8920842);
        double r8920875 = r8920840 / r8920874;
        double r8920876 = r8920873 / r8920875;
        double r8920877 = r8920840 / r8920837;
        double r8920878 = r8920875 * r8920875;
        double r8920879 = cbrt(r8920848);
        double r8920880 = r8920878 * r8920879;
        double r8920881 = r8920877 / r8920880;
        double r8920882 = r8920876 * r8920881;
        double r8920883 = r8920871 * r8920882;
        double r8920884 = r8920855 ? r8920866 : r8920883;
        double r8920885 = r8920832 ? r8920853 : r8920884;
        return r8920885;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -1.079547548008425e-56

   1. Initial program 22.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 9.7

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{t}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 9.7

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{t}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \sin k}}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   5. Applied times-frac 7.7

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{t}{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\sin k}}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   6. Applied *-un-lft-identity 7.7

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{\color{blue}{1 \cdot t}}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   7. Applied times-frac 6.9

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\frac{1}{\frac{\frac{\ell}{t}}{1}} \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}\right)} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   8. Applied associate-*l* 6.4

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\frac{1}{\frac{\frac{\ell}{t}}{1}} \cdot \left(\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)\right)}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 6.6

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

   11. Applied add-sqr-sqrt 6.7

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

   12. Applied times-frac 6.7

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

   13. Applied times-frac 3.5

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

   14. Simplified 3.5

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

   if -1.079547548008425e-56 < t < 8.337567466703915e+16

   1. Initial program 49.7

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 36.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{t}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 36.1

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{t}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \sin k}}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   5. Applied times-frac 34.2

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{t}{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\sin k}}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   6. Applied *-un-lft-identity 34.2

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{\color{blue}{1 \cdot t}}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   7. Applied times-frac 34.2

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\frac{1}{\frac{\frac{\ell}{t}}{1}} \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}\right)} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   8. Applied associate-*l* 31.2

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\frac{1}{\frac{\frac{\ell}{t}}{1}} \cdot \left(\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)\right)}}\]

   9. Taylor expanded around inf 18.4

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{1}{\frac{\frac{\ell}{t}}{1}} \cdot \color{blue}{\left(\frac{\sin k \cdot {k}^{2}}{\ell} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell}\right)}}\]

   10. Simplified 15.6

       \[\leadsto \frac{\frac{2}{\tan k}}{\frac{1}{\frac{\frac{\ell}{t}}{1}} \cdot \color{blue}{\left(\frac{\sin k}{\frac{\ell}{k \cdot k}} + \frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot 2\right)}}\]

   if 8.337567466703915e+16 < t

   1. Initial program 21.2

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 7.5

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{t}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 7.5

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{t}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \sin k}}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   5. Applied times-frac 6.2

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{t}{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\sin k}}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   6. Applied *-un-lft-identity 6.2

      \[\leadsto \frac{\frac{2}{\tan k}}{\frac{\color{blue}{1 \cdot t}}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   7. Applied times-frac 5.4

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\frac{1}{\frac{\frac{\ell}{t}}{1}} \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}\right)} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   8. Applied associate-*l* 5.2

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\frac{1}{\frac{\frac{\ell}{t}}{1}} \cdot \left(\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)\right)}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 5.4

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

   11. Applied *-un-lft-identity 5.4

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

   12. Applied times-frac 5.4

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

   13. Applied times-frac 1.9

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

   14. Simplified 1.9

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\sqrt[3]{\tan k}}}{\sqrt[3]{\tan k}}}{\frac{1}{\frac{\ell}{t}}}} \cdot \frac{\frac{2}{\sqrt[3]{\tan k}}}{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]
   15. Using strategy rm

   16. Applied add-cube-cbrt 2.0

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

   17. Applied add-cube-cbrt 2.0

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

   18. Applied times-frac 2.0

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

   19. Applied tan-quot 2.0

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{\tan k}}}{\sqrt[3]{\color{blue}{\frac{\sin k}{\cos k}}}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\ell}{t}}}} \cdot \frac{\frac{2}{\sqrt[3]{\tan k}}}{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   20. Applied cbrt-div 2.0

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{\tan k}}}{\color{blue}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\cos k}}}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\ell}{t}}}} \cdot \frac{\frac{2}{\sqrt[3]{\tan k}}}{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   21. Applied associate-/r/ 2.0

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{\tan k}}}{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\cos k}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\ell}{t}}}} \cdot \frac{\frac{2}{\sqrt[3]{\tan k}}}{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   22. Applied times-frac 1.8

       \[\leadsto \color{blue}{\left(\frac{\frac{\frac{1}{\sqrt[3]{\tan k}}}{\sqrt[3]{\sin k}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\ell}{t}}}}\right)} \cdot \frac{\frac{2}{\sqrt[3]{\tan k}}}{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   23. Simplified 1.8

       \[\leadsto \left(\color{blue}{\frac{\frac{1}{\sqrt[3]{\tan k}}}{\left(\frac{1}{\sqrt[3]{\frac{\ell}{t}}} \cdot \frac{1}{\sqrt[3]{\frac{\ell}{t}}}\right) \cdot \sqrt[3]{\sin k}}} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\ell}{t}}}}\right) \cdot \frac{\frac{2}{\sqrt[3]{\tan k}}}{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]

   24. Simplified 1.8

       \[\leadsto \left(\frac{\frac{1}{\sqrt[3]{\tan k}}}{\left(\frac{1}{\sqrt[3]{\frac{\ell}{t}}} \cdot \frac{1}{\sqrt[3]{\frac{\ell}{t}}}\right) \cdot \sqrt[3]{\sin k}} \cdot \color{blue}{\frac{\sqrt[3]{\cos k}}{\frac{1}{\sqrt[3]{\frac{\ell}{t}}}}}\right) \cdot \frac{\frac{2}{\sqrt[3]{\tan k}}}{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\]
   25. Using strategy rm

   26. Applied add-cube-cbrt 1.9

       \[\leadsto \left(\frac{\frac{1}{\sqrt[3]{\tan k}}}{\left(\frac{1}{\sqrt[3]{\frac{\ell}{t}}} \cdot \frac{1}{\sqrt[3]{\frac{\ell}{t}}}\right) \cdot \sqrt[3]{\sin k}} \cdot \frac{\sqrt[3]{\cos k}}{\frac{1}{\sqrt[3]{\frac{\ell}{t}}}}\right) \cdot \frac{\frac{2}{\sqrt[3]{\tan k}}}{\color{blue}{\left(\sqrt[3]{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)} \cdot \sqrt[3]{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}\right) \cdot \sqrt[3]{\frac{t}{\frac{\frac{\ell}{t}}{\sin k}} \cdot \mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.079547548008425 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{\tan k}}}{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}}\\ \mathbf{elif}\;t \le 8.337567466703915 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\left(\frac{\sin k}{\frac{\ell}{k \cdot k}} + \frac{t \cdot t}{\frac{\ell}{\sin k}} \cdot 2\right) \cdot \frac{1}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt[3]{\tan k}}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}} \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{t}{\frac{\frac{\ell}{t}}{\sin k}}}\right)} \cdot \left(\frac{\sqrt[3]{\cos k}}{\frac{1}{\sqrt[3]{\frac{\ell}{t}}}} \cdot \frac{\frac{1}{\sqrt[3]{\tan k}}}{\left(\frac{1}{\sqrt[3]{\frac{\ell}{t}}} \cdot \frac{1}{\sqrt[3]{\frac{\ell}{t}}}\right) \cdot \sqrt[3]{\sin k}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))