Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 12.0

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le 6.53962801490363253 \cdot 10^{-170}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \le 2.1181975674542142 \cdot 10^{146}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\ell}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r307667 = 2.0;
        double r307668 = t;
        double r307669 = 3.0;
        double r307670 = pow(r307668, r307669);
        double r307671 = l;
        double r307672 = r307671 * r307671;
        double r307673 = r307670 / r307672;
        double r307674 = k;
        double r307675 = sin(r307674);
        double r307676 = r307673 * r307675;
        double r307677 = tan(r307674);
        double r307678 = r307676 * r307677;
        double r307679 = 1.0;
        double r307680 = r307674 / r307668;
        double r307681 = pow(r307680, r307667);
        double r307682 = r307679 + r307681;
        double r307683 = r307682 - r307679;
        double r307684 = r307678 * r307683;
        double r307685 = r307667 / r307684;
        return r307685;
}

↓

double f(double t, double l, double k) {
        double r307686 = l;
        double r307687 = 6.5396280149036325e-170;
        bool r307688 = r307686 <= r307687;
        double r307689 = 2.0;
        double r307690 = 1.0;
        double r307691 = k;
        double r307692 = 2.0;
        double r307693 = r307689 / r307692;
        double r307694 = pow(r307691, r307693);
        double r307695 = t;
        double r307696 = 1.0;
        double r307697 = pow(r307695, r307696);
        double r307698 = r307694 * r307697;
        double r307699 = r307694 * r307698;
        double r307700 = r307690 / r307699;
        double r307701 = pow(r307700, r307696);
        double r307702 = cos(r307691);
        double r307703 = cbrt(r307702);
        double r307704 = r307703 * r307703;
        double r307705 = sin(r307691);
        double r307706 = cbrt(r307705);
        double r307707 = 4.0;
        double r307708 = pow(r307706, r307707);
        double r307709 = r307708 / r307686;
        double r307710 = cbrt(r307709);
        double r307711 = r307710 * r307710;
        double r307712 = cbrt(r307686);
        double r307713 = r307712 * r307712;
        double r307714 = r307711 / r307713;
        double r307715 = r307704 / r307714;
        double r307716 = pow(r307706, r307692);
        double r307717 = cbrt(r307716);
        double r307718 = r307717 * r307717;
        double r307719 = r307715 / r307718;
        double r307720 = r307701 * r307719;
        double r307721 = r307710 / r307712;
        double r307722 = r307703 / r307721;
        double r307723 = r307722 / r307717;
        double r307724 = r307720 * r307723;
        double r307725 = r307689 * r307724;
        double r307726 = 2.1181975674542142e+146;
        bool r307727 = r307686 <= r307726;
        double r307728 = cbrt(r307690);
        double r307729 = r307728 * r307728;
        double r307730 = r307729 / r307694;
        double r307731 = pow(r307730, r307696);
        double r307732 = r307728 / r307698;
        double r307733 = pow(r307732, r307696);
        double r307734 = pow(r307686, r307692);
        double r307735 = r307702 * r307734;
        double r307736 = pow(r307705, r307692);
        double r307737 = r307735 / r307736;
        double r307738 = r307733 * r307737;
        double r307739 = r307731 * r307738;
        double r307740 = r307689 * r307739;
        double r307741 = r307702 / r307709;
        double r307742 = r307741 / r307718;
        double r307743 = r307701 * r307742;
        double r307744 = r307686 / r307717;
        double r307745 = r307743 * r307744;
        double r307746 = r307689 * r307745;
        double r307747 = r307727 ? r307740 : r307746;
        double r307748 = r307688 ? r307725 : r307747;
        return r307748;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < 6.5396280149036325e-170

   1. Initial program 47.7

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

   2. Simplified 39.8

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

   3. Taylor expanded around inf 21.6

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

   5. Applied sqr-pow 21.6

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

   6. Applied associate-*l* 20.1

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

   8. Applied add-cube-cbrt 20.3

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

   9. Applied unpow-prod-down 20.3

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

   10. Applied associate-/r* 20.1

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

   11. Simplified 17.4

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

   13. Applied add-cube-cbrt 17.4

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

   14. Applied add-cube-cbrt 17.4

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

   15. Applied add-cube-cbrt 17.5

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

   16. Applied times-frac 17.5

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

   17. Applied add-cube-cbrt 17.5

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

   18. Applied times-frac 17.2

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

   19. Applied times-frac 16.0

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

   20. Applied associate-*r* 12.0

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

   if 6.5396280149036325e-170 < l < 2.1181975674542142e+146

   1. Initial program 44.8

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

   2. Simplified 34.8

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

   3. Taylor expanded around inf 11.1

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

   5. Applied sqr-pow 11.1

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

   6. Applied associate-*l* 7.2

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

   8. Applied add-cube-cbrt 7.2

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

   9. Applied times-frac 6.8

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

   10. Applied unpow-prod-down 6.8

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

   11. Applied associate-*l* 3.8

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

   if 2.1181975674542142e+146 < l

   1. Initial program 63.6

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

   2. Simplified 63.2

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

   3. Taylor expanded around inf 62.6

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

   5. Applied sqr-pow 62.6

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

   6. Applied associate-*l* 62.2

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

   8. Applied add-cube-cbrt 62.2

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

   9. Applied unpow-prod-down 62.2

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

   10. Applied associate-/r* 62.2

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

   11. Simplified 62.2

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

   13. Applied add-cube-cbrt 62.2

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

   14. Applied associate-/r/ 62.2

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

   15. Applied times-frac 62.2

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

   16. Applied associate-*r* 38.5

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 6.53962801490363253 \cdot 10^{-170}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \le 2.1181975674542142 \cdot 10^{146}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\ell}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))))