Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 10.8

Time: 2.9m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{4}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{4}}}{\sqrt[3]{\ell}}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 1.52399384553112956262300731935669632006 \cdot 10^{193}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\ell}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r334818 = 2.0;
        double r334819 = t;
        double r334820 = 3.0;
        double r334821 = pow(r334819, r334820);
        double r334822 = l;
        double r334823 = r334822 * r334822;
        double r334824 = r334821 / r334823;
        double r334825 = k;
        double r334826 = sin(r334825);
        double r334827 = r334824 * r334826;
        double r334828 = tan(r334825);
        double r334829 = r334827 * r334828;
        double r334830 = 1.0;
        double r334831 = r334825 / r334819;
        double r334832 = pow(r334831, r334818);
        double r334833 = r334830 + r334832;
        double r334834 = r334833 - r334830;
        double r334835 = r334829 * r334834;
        double r334836 = r334818 / r334835;
        return r334836;
}

↓

double f(double t, double l, double k) {
        double r334837 = l;
        double r334838 = r334837 * r334837;
        double r334839 = 0.0;
        bool r334840 = r334838 <= r334839;
        double r334841 = 2.0;
        double r334842 = 1.0;
        double r334843 = k;
        double r334844 = 2.0;
        double r334845 = r334841 / r334844;
        double r334846 = pow(r334843, r334845);
        double r334847 = t;
        double r334848 = 1.0;
        double r334849 = pow(r334847, r334848);
        double r334850 = r334846 * r334849;
        double r334851 = r334846 * r334850;
        double r334852 = r334842 / r334851;
        double r334853 = pow(r334852, r334848);
        double r334854 = cos(r334843);
        double r334855 = cbrt(r334854);
        double r334856 = r334855 * r334855;
        double r334857 = sin(r334843);
        double r334858 = cbrt(r334857);
        double r334859 = 4.0;
        double r334860 = pow(r334858, r334859);
        double r334861 = sqrt(r334860);
        double r334862 = cbrt(r334837);
        double r334863 = r334862 * r334862;
        double r334864 = r334861 / r334863;
        double r334865 = r334864 / r334842;
        double r334866 = r334856 / r334865;
        double r334867 = cbrt(r334842);
        double r334868 = pow(r334867, r334844);
        double r334869 = r334866 / r334868;
        double r334870 = r334853 * r334869;
        double r334871 = r334861 / r334862;
        double r334872 = r334871 / r334837;
        double r334873 = r334855 / r334872;
        double r334874 = pow(r334858, r334844);
        double r334875 = r334873 / r334874;
        double r334876 = r334870 * r334875;
        double r334877 = r334841 * r334876;
        double r334878 = 1.5239938455311296e+193;
        bool r334879 = r334838 <= r334878;
        double r334880 = sqrt(r334842);
        double r334881 = r334880 / r334846;
        double r334882 = pow(r334881, r334848);
        double r334883 = r334880 / r334850;
        double r334884 = pow(r334883, r334848);
        double r334885 = pow(r334837, r334844);
        double r334886 = r334854 * r334885;
        double r334887 = pow(r334857, r334844);
        double r334888 = r334886 / r334887;
        double r334889 = r334884 * r334888;
        double r334890 = r334882 * r334889;
        double r334891 = r334841 * r334890;
        double r334892 = r334860 / r334837;
        double r334893 = r334854 / r334892;
        double r334894 = sqrt(r334874);
        double r334895 = r334893 / r334894;
        double r334896 = r334853 * r334895;
        double r334897 = r334837 / r334894;
        double r334898 = r334896 * r334897;
        double r334899 = r334841 * r334898;
        double r334900 = r334879 ? r334891 : r334899;
        double r334901 = r334840 ? r334877 : r334900;
        return r334901;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 0.0

   1. Initial program 46.4

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

   2. Simplified 37.5

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \left({t}^{3} \cdot \tan k\right)\right) \cdot \sin k}}\]

   3. Taylor expanded around inf 19.9

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 19.9

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

   6. Applied associate-*l* 19.9

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 19.9

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

   9. Applied unpow-prod-down 19.9

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

   10. Applied associate-/r* 19.8

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

   11. Simplified 13.9

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
   12. Using strategy rm

   13. Applied *-un-lft-identity 13.9

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

   14. Applied cbrt-prod 13.9

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

   15. Applied unpow-prod-down 13.9

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

   16. Applied *-un-lft-identity 13.9

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

   17. Applied add-cube-cbrt 13.9

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

   18. Applied add-sqr-sqrt 13.9

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

   19. Applied times-frac 13.9

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

   20. Applied times-frac 13.9

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

   21. Applied add-cube-cbrt 13.9

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

   22. Applied times-frac 13.5

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

   23. Applied times-frac 11.1

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

   24. Applied associate-*r* 8.0

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

   if 0.0 < (* l l) < 1.5239938455311296e+193

   1. Initial program 44.6

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

   2. Simplified 34.5

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \left({t}^{3} \cdot \tan k\right)\right) \cdot \sin k}}\]

   3. Taylor expanded around inf 8.2

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 8.2

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

   6. Applied associate-*l* 4.5

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
   7. Using strategy rm

   8. Applied add-sqr-sqrt 4.5

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

   9. Applied times-frac 4.3

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

   10. Applied unpow-prod-down 4.3

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

   11. Applied associate-*l* 2.3

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

   if 1.5239938455311296e+193 < (* l l)

   1. Initial program 59.3

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

   2. Simplified 56.8

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{2} \cdot \left({t}^{3} \cdot \tan k\right)\right) \cdot \sin k}}\]

   3. Taylor expanded around inf 51.4

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 51.4

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

   6. Applied associate-*l* 49.6

      \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 49.8

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

   9. Applied unpow-prod-down 49.8

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

   10. Applied associate-/r* 49.8

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

   11. Simplified 49.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt 49.8

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

   14. Applied associate-/r/ 49.8

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

   15. Applied times-frac 49.8

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

   16. Applied associate-*r* 32.7

       \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\ell}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 0.0:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{4}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{4}}}{\sqrt[3]{\ell}}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 1.52399384553112956262300731935669632006 \cdot 10^{193}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\ell}{\sqrt{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))