Average Error: 48.8 → 10.6
Time: 4.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -3.2121767733629187 \cdot 10^{153}:\\ \;\;\;\;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{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le -4.46760789925130493 \cdot 10^{-137}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{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{elif}\;\ell \le 3.6473195150792043 \cdot 10^{-200}:\\ \;\;\;\;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{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k}}\right)\\ \mathbf{elif}\;\ell \le 1.3529385316824283 \cdot 10^{146}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{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 9.37568809863626905 \cdot 10^{197}:\\ \;\;\;\;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{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\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]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot e^{\left(\log 1 - \left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)\right) \cdot 1 + \left(\left(\log \left(\cos k\right) - \left(\log \left(\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)\right)}\\ \end{array}\]
\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 -3.2121767733629187 \cdot 10^{153}:\\
\;\;\;\;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{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\

\mathbf{elif}\;\ell \le -4.46760789925130493 \cdot 10^{-137}:\\
\;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{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{elif}\;\ell \le 3.6473195150792043 \cdot 10^{-200}:\\
\;\;\;\;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{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k}}\right)\\

\mathbf{elif}\;\ell \le 1.3529385316824283 \cdot 10^{146}:\\
\;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{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 9.37568809863626905 \cdot 10^{197}:\\
\;\;\;\;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{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\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]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot e^{\left(\log 1 - \left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)\right) \cdot 1 + \left(\left(\log \left(\cos k\right) - \left(\log \left(\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)\right)}\\

\end{array}
double f(double t, double l, double k) {
        double r347616 = 2.0;
        double r347617 = t;
        double r347618 = 3.0;
        double r347619 = pow(r347617, r347618);
        double r347620 = l;
        double r347621 = r347620 * r347620;
        double r347622 = r347619 / r347621;
        double r347623 = k;
        double r347624 = sin(r347623);
        double r347625 = r347622 * r347624;
        double r347626 = tan(r347623);
        double r347627 = r347625 * r347626;
        double r347628 = 1.0;
        double r347629 = r347623 / r347617;
        double r347630 = pow(r347629, r347616);
        double r347631 = r347628 + r347630;
        double r347632 = r347631 - r347628;
        double r347633 = r347627 * r347632;
        double r347634 = r347616 / r347633;
        return r347634;
}
↓
double f(double t, double l, double k) {
        double r347635 = l;
        double r347636 = -3.2121767733629187e+153;
        bool r347637 = r347635 <= r347636;
        double r347638 = 2.0;
        double r347639 = 1.0;
        double r347640 = k;
        double r347641 = 2.0;
        double r347642 = r347638 / r347641;
        double r347643 = pow(r347640, r347642);
        double r347644 = t;
        double r347645 = 1.0;
        double r347646 = pow(r347644, r347645);
        double r347647 = r347643 * r347646;
        double r347648 = r347643 * r347647;
        double r347649 = r347639 / r347648;
        double r347650 = pow(r347649, r347645);
        double r347651 = sin(r347640);
        double r347652 = cbrt(r347651);
        double r347653 = 4.0;
        double r347654 = pow(r347652, r347653);
        double r347655 = r347654 / r347635;
        double r347656 = r347639 / r347655;
        double r347657 = r347652 * r347652;
        double r347658 = cbrt(r347657);
        double r347659 = pow(r347658, r347641);
        double r347660 = r347656 / r347659;
        double r347661 = r347650 * r347660;
        double r347662 = cos(r347640);
        double r347663 = r347639 / r347635;
        double r347664 = r347662 / r347663;
        double r347665 = cbrt(r347652);
        double r347666 = pow(r347665, r347641);
        double r347667 = r347664 / r347666;
        double r347668 = r347661 * r347667;
        double r347669 = r347638 * r347668;
        double r347670 = -4.467607899251305e-137;
        bool r347671 = r347635 <= r347670;
        double r347672 = sqrt(r347639);
        double r347673 = r347672 / r347643;
        double r347674 = pow(r347673, r347645);
        double r347675 = r347639 / r347647;
        double r347676 = pow(r347675, r347645);
        double r347677 = pow(r347635, r347641);
        double r347678 = r347662 * r347677;
        double r347679 = pow(r347651, r347641);
        double r347680 = r347678 / r347679;
        double r347681 = r347676 * r347680;
        double r347682 = r347674 * r347681;
        double r347683 = r347638 * r347682;
        double r347684 = 3.647319515079204e-200;
        bool r347685 = r347635 <= r347684;
        double r347686 = cbrt(r347635);
        double r347687 = r347686 * r347686;
        double r347688 = r347639 / r347687;
        double r347689 = r347688 / r347639;
        double r347690 = r347639 / r347689;
        double r347691 = r347690 / r347652;
        double r347692 = r347650 * r347691;
        double r347693 = r347654 / r347686;
        double r347694 = r347693 / r347635;
        double r347695 = r347662 / r347694;
        double r347696 = r347695 / r347652;
        double r347697 = r347692 * r347696;
        double r347698 = r347638 * r347697;
        double r347699 = 1.3529385316824283e+146;
        bool r347700 = r347635 <= r347699;
        double r347701 = r347639 / r347643;
        double r347702 = pow(r347701, r347645);
        double r347703 = r347655 / r347635;
        double r347704 = r347662 / r347703;
        double r347705 = pow(r347652, r347641);
        double r347706 = r347704 / r347705;
        double r347707 = r347676 * r347706;
        double r347708 = r347702 * r347707;
        double r347709 = r347638 * r347708;
        double r347710 = 9.375688098636269e+197;
        bool r347711 = r347635 <= r347710;
        double r347712 = cbrt(r347662);
        double r347713 = r347712 * r347712;
        double r347714 = pow(r347658, r347653);
        double r347715 = r347714 / r347687;
        double r347716 = r347715 / r347687;
        double r347717 = r347713 / r347716;
        double r347718 = r347665 * r347665;
        double r347719 = pow(r347718, r347641);
        double r347720 = r347717 / r347719;
        double r347721 = r347650 * r347720;
        double r347722 = pow(r347665, r347653);
        double r347723 = r347722 / r347686;
        double r347724 = r347723 / r347686;
        double r347725 = r347712 / r347724;
        double r347726 = r347725 / r347666;
        double r347727 = r347721 * r347726;
        double r347728 = r347638 * r347727;
        double r347729 = log(r347639);
        double r347730 = log(r347643);
        double r347731 = log(r347640);
        double r347732 = r347731 * r347642;
        double r347733 = log(r347644);
        double r347734 = r347733 * r347645;
        double r347735 = r347732 + r347734;
        double r347736 = r347730 + r347735;
        double r347737 = r347729 - r347736;
        double r347738 = r347737 * r347645;
        double r347739 = log(r347662);
        double r347740 = log(r347655);
        double r347741 = log(r347635);
        double r347742 = r347740 - r347741;
        double r347743 = r347739 - r347742;
        double r347744 = log(r347705);
        double r347745 = r347743 - r347744;
        double r347746 = r347738 + r347745;
        double r347747 = exp(r347746);
        double r347748 = r347638 * r347747;
        double r347749 = r347711 ? r347728 : r347748;
        double r347750 = r347700 ? r347709 : r347749;
        double r347751 = r347685 ? r347698 : r347750;
        double r347752 = r347671 ? r347683 : r347751;
        double r347753 = r347637 ? r347669 : r347752;
        return r347753;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 6 regimes

  2. if l < -3.2121767733629187e+153

    1. Initial program 64.0

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

    2. Simplified63.9

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

    3. Taylor expanded around inf 63.8

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

    5. Applied sqr-pow63.8

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

    6. Applied associate-*l*63.8

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

    8. Applied add-cube-cbrt63.8

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

    9. Applied unpow-prod-down63.8

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

    10. Applied associate-/r*63.8

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

    11. Simplified63.8

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

    13. Applied add-cube-cbrt63.8

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

    14. Applied cbrt-prod63.8

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

    15. Applied unpow-prod-down63.8

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

    16. Applied div-inv63.8

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

    17. Applied *-un-lft-identity63.8

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

    18. Applied times-frac63.8

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

    19. Applied times-frac63.8

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

    20. Applied associate-*r*42.0

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

    if -3.2121767733629187e+153 < l < -4.467607899251305e-137

    1. Initial program 45.4

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

    2. Simplified36.0

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

    3. Taylor expanded around inf 12.1

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

    5. Applied sqr-pow12.1

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

    6. Applied associate-*l*7.8

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

    8. Applied add-sqr-sqrt7.8

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

    9. Applied times-frac7.4

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

    10. Applied unpow-prod-down7.4

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

    11. Applied associate-*l*4.2

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

    12. Simplified4.2

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

    if -4.467607899251305e-137 < l < 3.647319515079204e-200

    1. Initial program 47.0

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

    2. Simplified38.1

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

    3. Taylor expanded around inf 18.4

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

    5. Applied sqr-pow18.4

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

    6. Applied associate-*l*18.4

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

    8. Applied add-cube-cbrt18.4

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

    9. Applied unpow-prod-down18.4

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

    10. Applied associate-/r*18.2

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

    11. Simplified13.2

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

    13. Applied unpow213.2

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

    14. Applied *-un-lft-identity13.2

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

    15. Applied add-cube-cbrt13.2

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

    16. Applied *-un-lft-identity13.2

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

    17. Applied unpow-prod-down13.2

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

    18. Applied times-frac13.2

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

    19. Applied times-frac13.2

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

    20. Applied *-un-lft-identity13.2

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

    21. Applied times-frac12.8

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

    22. Applied times-frac10.4

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

    23. Applied associate-*r*7.3

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

    24. Simplified7.3

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

    if 3.647319515079204e-200 < l < 1.3529385316824283e+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. Simplified36.0

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

    3. Taylor expanded around inf 13.0

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

    5. Applied sqr-pow13.0

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

    6. Applied associate-*l*9.3

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

    8. Applied add-cube-cbrt9.9

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

    9. Applied unpow-prod-down9.9

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

    10. Applied associate-/r*9.5

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

    11. Simplified7.9

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

    13. Applied *-un-lft-identity7.9

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

    14. Applied times-frac7.6

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

    15. Applied unpow-prod-down7.6

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

    16. Applied associate-*l*4.9

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

    if 1.3529385316824283e+146 < l < 9.375688098636269e+197

    1. Initial program 61.9

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

    2. Simplified61.8

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

    3. Taylor expanded around inf 60.1

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

    5. Applied sqr-pow60.1

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

    6. Applied associate-*l*58.8

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

    8. Applied add-cube-cbrt58.8

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

    9. Applied unpow-prod-down58.8

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

    10. Applied associate-/r*58.8

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

    11. Simplified58.8

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

    13. Applied add-cube-cbrt58.9

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

    14. Applied unpow-prod-down58.9

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

    15. Applied add-cube-cbrt58.9

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

    16. Applied add-cube-cbrt58.9

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

    17. Applied add-cube-cbrt58.9

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

    18. Applied cbrt-prod58.9

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

    19. Applied unpow-prod-down58.9

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

    20. Applied times-frac58.9

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

    21. Applied times-frac58.9

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

    22. Applied add-cube-cbrt58.9

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

    23. Applied times-frac58.9

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

    24. Applied times-frac58.9

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

    25. Applied associate-*r*26.0

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

    if 9.375688098636269e+197 < l

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around inf 64.0

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

    5. Applied sqr-pow64.0

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

    6. Applied associate-*l*64.0

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

    8. Applied add-cube-cbrt64.0

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

    9. Applied unpow-prod-down64.0

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

    10. Applied associate-/r*64.0

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

    11. Simplified64.0

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

    13. Applied add-exp-log64.0

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

    14. Applied add-exp-log64.0

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

    15. Applied add-exp-log64.0

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

    16. Applied div-exp64.0

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

    17. Applied add-exp-log64.0

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

    18. Applied div-exp64.0

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

    19. Applied div-exp64.0

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

    20. Applied add-exp-log64.0

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

    21. Applied pow-exp64.0

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

    22. Applied add-exp-log64.0

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

    23. Applied pow-exp64.0

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

    24. Applied prod-exp64.0

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

    25. Applied add-exp-log64.0

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

    26. Applied prod-exp64.0

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

    27. Applied add-exp-log64.0

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

    28. Applied div-exp64.0

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

    29. Applied pow-exp64.0

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

    30. Applied prod-exp36.6

      \[\leadsto 2 \cdot \color{blue}{e^{\left(\log 1 - \left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)\right) \cdot 1 + \left(\left(\log \left(\cos k\right) - \left(\log \left(\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)\right)}}\]
  3. Recombined 6 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.2121767733629187 \cdot 10^{153}:\\ \;\;\;\;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{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le -4.46760789925130493 \cdot 10^{-137}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{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{elif}\;\ell \le 3.6473195150792043 \cdot 10^{-200}:\\ \;\;\;\;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{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt[3]{\ell}}}{\ell}}}{\sqrt[3]{\sin k}}\right)\\ \mathbf{elif}\;\ell \le 1.3529385316824283 \cdot 10^{146}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{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 9.37568809863626905 \cdot 10^{197}:\\ \;\;\;\;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{{\left(\sqrt[3]{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\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]{\sqrt[3]{\sin k}}\right)}^{4}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot e^{\left(\log 1 - \left(\log \left({k}^{\left(\frac{2}{2}\right)}\right) + \left(\log k \cdot \frac{2}{2} + \log t \cdot 1\right)\right)\right) \cdot 1 + \left(\left(\log \left(\cos k\right) - \left(\log \left(\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}\right) - \log \ell\right)\right) - \log \left({\left(\sqrt[3]{\sin k}\right)}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))