Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 11.1

Time: 1.6m

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 -9.188359687395318902854474088693230583334 \cdot 10^{130}:\\ \;\;\;\;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)\\ \mathbf{elif}\;\ell \le -1.720175503403893340383013706398982584406 \cdot 10^{-227}:\\ \;\;\;\;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{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 2.174121878485278959660471774596909385214 \cdot 10^{-187}:\\ \;\;\;\;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{1}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{1}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\ell}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le 5.502176481489900707136350807955780337278 \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{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\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{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt{\ell}}}{\sqrt{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r307631 = 2.0;
        double r307632 = t;
        double r307633 = 3.0;
        double r307634 = pow(r307632, r307633);
        double r307635 = l;
        double r307636 = r307635 * r307635;
        double r307637 = r307634 / r307636;
        double r307638 = k;
        double r307639 = sin(r307638);
        double r307640 = r307637 * r307639;
        double r307641 = tan(r307638);
        double r307642 = r307640 * r307641;
        double r307643 = 1.0;
        double r307644 = r307638 / r307632;
        double r307645 = pow(r307644, r307631);
        double r307646 = r307643 + r307645;
        double r307647 = r307646 - r307643;
        double r307648 = r307642 * r307647;
        double r307649 = r307631 / r307648;
        return r307649;
}

↓

double f(double t, double l, double k) {
        double r307650 = l;
        double r307651 = -9.188359687395319e+130;
        bool r307652 = r307650 <= r307651;
        double r307653 = 2.0;
        double r307654 = 1.0;
        double r307655 = k;
        double r307656 = 2.0;
        double r307657 = r307653 / r307656;
        double r307658 = pow(r307655, r307657);
        double r307659 = t;
        double r307660 = 1.0;
        double r307661 = pow(r307659, r307660);
        double r307662 = r307658 * r307661;
        double r307663 = r307658 * r307662;
        double r307664 = r307654 / r307663;
        double r307665 = pow(r307664, r307660);
        double r307666 = cos(r307655);
        double r307667 = sin(r307655);
        double r307668 = cbrt(r307667);
        double r307669 = 4.0;
        double r307670 = pow(r307668, r307669);
        double r307671 = r307670 / r307650;
        double r307672 = r307666 / r307671;
        double r307673 = pow(r307668, r307656);
        double r307674 = cbrt(r307673);
        double r307675 = r307674 * r307674;
        double r307676 = r307672 / r307675;
        double r307677 = r307665 * r307676;
        double r307678 = r307650 / r307674;
        double r307679 = r307677 * r307678;
        double r307680 = r307653 * r307679;
        double r307681 = -1.7201755034038933e-227;
        bool r307682 = r307650 <= r307681;
        double r307683 = cbrt(r307654);
        double r307684 = r307683 * r307683;
        double r307685 = r307684 / r307658;
        double r307686 = pow(r307685, r307660);
        double r307687 = r307683 / r307662;
        double r307688 = pow(r307687, r307660);
        double r307689 = r307671 / r307650;
        double r307690 = r307666 / r307689;
        double r307691 = r307690 / r307673;
        double r307692 = r307688 * r307691;
        double r307693 = r307686 * r307692;
        double r307694 = r307653 * r307693;
        double r307695 = 2.174121878485279e-187;
        bool r307696 = r307650 <= r307695;
        double r307697 = r307673 / r307654;
        double r307698 = cbrt(r307650);
        double r307699 = r307698 * r307698;
        double r307700 = r307697 / r307699;
        double r307701 = r307654 / r307700;
        double r307702 = r307701 / r307654;
        double r307703 = r307665 * r307702;
        double r307704 = r307669 / r307656;
        double r307705 = pow(r307668, r307704);
        double r307706 = r307705 / r307650;
        double r307707 = r307706 / r307698;
        double r307708 = r307666 / r307707;
        double r307709 = r307708 / r307673;
        double r307710 = r307703 * r307709;
        double r307711 = r307653 * r307710;
        double r307712 = 5.502176481489901e+146;
        bool r307713 = r307650 <= r307712;
        double r307714 = cbrt(r307666);
        double r307715 = r307714 * r307714;
        double r307716 = sqrt(r307650);
        double r307717 = r307654 / r307716;
        double r307718 = r307717 / r307716;
        double r307719 = r307715 / r307718;
        double r307720 = cbrt(r307668);
        double r307721 = r307720 * r307720;
        double r307722 = pow(r307721, r307656);
        double r307723 = r307719 / r307722;
        double r307724 = r307665 * r307723;
        double r307725 = r307670 / r307716;
        double r307726 = r307725 / r307716;
        double r307727 = r307714 / r307726;
        double r307728 = pow(r307720, r307656);
        double r307729 = r307727 / r307728;
        double r307730 = r307724 * r307729;
        double r307731 = r307653 * r307730;
        double r307732 = r307713 ? r307694 : r307731;
        double r307733 = r307696 ? r307711 : r307732;
        double r307734 = r307682 ? r307694 : r307733;
        double r307735 = r307652 ? r307680 : r307734;
        return r307735;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if l < -9.188359687395319e+130

   1. Initial program 62.1

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

   2. Simplified 61.1

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

   3. Taylor expanded around inf 59.5

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

   5. Applied sqr-pow 59.5

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

   6. Applied associate-*l* 58.1

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

   8. Applied add-cube-cbrt 58.2

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

   9. Applied unpow-prod-down 58.2

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

   10. Applied associate-/r* 58.2

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

   11. Simplified 58.2

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

   13. Applied add-cube-cbrt 58.2

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

   14. Applied associate-/r/ 58.2

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

   15. Applied times-frac 58.2

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

   16. Applied associate-*r* 36.4

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

   if -9.188359687395319e+130 < l < -1.7201755034038933e-227 or 2.174121878485279e-187 < l < 5.502176481489901e+146

   1. Initial program 45.5

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

   2. Simplified 35.8

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

   3. Taylor expanded around inf 11.7

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

   5. Applied sqr-pow 11.7

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

   6. Applied associate-*l* 8.7

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

   8. Applied add-cube-cbrt 9.3

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

   9. Applied unpow-prod-down 9.3

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

   10. Applied associate-/r* 9.0

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

   11. Simplified 7.2

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

   13. Applied add-cube-cbrt 7.2

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

   14. Applied times-frac 6.9

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

   15. Applied unpow-prod-down 6.9

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

   16. Applied associate-*l* 4.5

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

   if -1.7201755034038933e-227 < l < 2.174121878485279e-187

   1. Initial program 46.3

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

   2. Simplified 37.6

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

   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 15.8

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

   13. Applied *-un-lft-identity 15.8

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

   14. Applied unpow-prod-down 15.8

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

   15. Applied add-cube-cbrt 15.8

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

   16. Applied *-un-lft-identity 15.8

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

   17. Applied sqr-pow 15.8

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

   18. Applied times-frac 15.1

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

   19. Applied times-frac 15.1

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

   20. Applied *-un-lft-identity 15.1

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

   21. Applied times-frac 14.7

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

   22. Applied times-frac 11.7

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

   23. Applied associate-*r* 9.2

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

   24. Simplified 9.2

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

   if 5.502176481489901e+146 < l

   1. Initial program 63.3

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

   2. Simplified 63.0

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

   3. Taylor expanded around inf 62.0

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

   5. Applied sqr-pow 62.0

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

   6. Applied associate-*l* 61.6

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

   8. Applied add-cube-cbrt 61.6

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

   9. Applied unpow-prod-down 61.6

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

   10. Applied associate-/r* 61.6

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

   11. Simplified 61.6

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

   13. Applied add-cube-cbrt 61.6

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

   14. Applied unpow-prod-down 61.6

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

   15. Applied add-sqr-sqrt 61.6

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

   16. Applied add-sqr-sqrt 61.6

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

   17. Applied *-un-lft-identity 61.6

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

   18. Applied unpow-prod-down 61.6

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

   19. Applied times-frac 61.6

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

   20. Applied times-frac 61.6

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

   21. Applied add-cube-cbrt 61.6

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

   22. Applied times-frac 61.6

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

   23. Applied times-frac 61.6

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

   24. Applied associate-*r* 40.1

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

   25. Simplified 40.1

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -9.188359687395318902854474088693230583334 \cdot 10^{130}:\\ \;\;\;\;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)\\ \mathbf{elif}\;\ell \le -1.720175503403893340383013706398982584406 \cdot 10^{-227}:\\ \;\;\;\;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{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\right)\\ \mathbf{elif}\;\ell \le 2.174121878485278959660471774596909385214 \cdot 10^{-187}:\\ \;\;\;\;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{1}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{1}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\ell}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le 5.502176481489900707136350807955780337278 \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{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{{\left(\sqrt[3]{\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{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt{\ell}}}{\sqrt{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))