Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 13.4

Time: 57.6s

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 -2.385068763139975120500013847597821926732 \cdot 10^{91}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\frac{\sqrt[3]{\ell}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \mathbf{elif}\;\ell \le -1.305991081232837483759897853819755248256 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(\frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left(t \cdot \left(\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)\right)\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{\frac{\left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right)\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\ell \cdot \ell}}{\cos k}\right)}\\ \mathbf{elif}\;\ell \le 7.061293343570338284594492301637048497777 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot k}} + \frac{2}{\ell} \cdot \frac{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t}{\ell}}\\ \mathbf{elif}\;\ell \le 8815224158947631239444986233777815552:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(\frac{\frac{t \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\ell} \cdot \frac{\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)}{\ell}}{\cos k} + \left(\frac{t \cdot \left({\left(\sqrt[3]{-1}\right)}^{9} \cdot \left(t \cdot t\right)\right)}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{3}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r4951578 = 2.0;
        double r4951579 = t;
        double r4951580 = 3.0;
        double r4951581 = pow(r4951579, r4951580);
        double r4951582 = l;
        double r4951583 = r4951582 * r4951582;
        double r4951584 = r4951581 / r4951583;
        double r4951585 = k;
        double r4951586 = sin(r4951585);
        double r4951587 = r4951584 * r4951586;
        double r4951588 = tan(r4951585);
        double r4951589 = r4951587 * r4951588;
        double r4951590 = 1.0;
        double r4951591 = r4951585 / r4951579;
        double r4951592 = pow(r4951591, r4951578);
        double r4951593 = r4951590 + r4951592;
        double r4951594 = r4951593 + r4951590;
        double r4951595 = r4951589 * r4951594;
        double r4951596 = r4951578 / r4951595;
        return r4951596;
}

↓

double f(double t, double l, double k) {
        double r4951597 = l;
        double r4951598 = -2.385068763139975e+91;
        bool r4951599 = r4951597 <= r4951598;
        double r4951600 = 2.0;
        double r4951601 = k;
        double r4951602 = tan(r4951601);
        double r4951603 = sin(r4951601);
        double r4951604 = t;
        double r4951605 = cbrt(r4951604);
        double r4951606 = 3.0;
        double r4951607 = pow(r4951605, r4951606);
        double r4951608 = r4951607 / r4951597;
        double r4951609 = r4951603 * r4951608;
        double r4951610 = r4951605 * r4951605;
        double r4951611 = 2.0;
        double r4951612 = r4951606 / r4951611;
        double r4951613 = pow(r4951610, r4951612);
        double r4951614 = sqrt(r4951613);
        double r4951615 = cbrt(r4951597);
        double r4951616 = r4951612 / r4951611;
        double r4951617 = pow(r4951610, r4951616);
        double r4951618 = r4951615 / r4951617;
        double r4951619 = r4951614 / r4951618;
        double r4951620 = r4951609 * r4951619;
        double r4951621 = r4951615 * r4951615;
        double r4951622 = r4951621 / r4951617;
        double r4951623 = r4951614 / r4951622;
        double r4951624 = r4951620 * r4951623;
        double r4951625 = r4951602 * r4951624;
        double r4951626 = r4951601 / r4951604;
        double r4951627 = pow(r4951626, r4951600);
        double r4951628 = 1.0;
        double r4951629 = r4951627 + r4951628;
        double r4951630 = r4951629 + r4951628;
        double r4951631 = r4951625 * r4951630;
        double r4951632 = r4951600 / r4951631;
        double r4951633 = -1.3059910812328375e-155;
        bool r4951634 = r4951597 <= r4951633;
        double r4951635 = 1.0;
        double r4951636 = -1.0;
        double r4951637 = pow(r4951636, r4951606);
        double r4951638 = r4951635 / r4951637;
        double r4951639 = pow(r4951638, r4951628);
        double r4951640 = cbrt(r4951636);
        double r4951641 = 9.0;
        double r4951642 = pow(r4951640, r4951641);
        double r4951643 = r4951603 * r4951601;
        double r4951644 = r4951643 * r4951643;
        double r4951645 = r4951604 * r4951644;
        double r4951646 = r4951642 * r4951645;
        double r4951647 = r4951597 * r4951597;
        double r4951648 = cos(r4951601);
        double r4951649 = r4951647 * r4951648;
        double r4951650 = r4951646 / r4951649;
        double r4951651 = r4951603 * r4951604;
        double r4951652 = r4951651 * r4951651;
        double r4951653 = r4951604 * r4951652;
        double r4951654 = r4951653 * r4951642;
        double r4951655 = r4951654 / r4951647;
        double r4951656 = r4951655 / r4951648;
        double r4951657 = r4951600 * r4951656;
        double r4951658 = r4951650 + r4951657;
        double r4951659 = r4951639 * r4951658;
        double r4951660 = r4951600 / r4951659;
        double r4951661 = 7.061293343570338e-144;
        bool r4951662 = r4951597 <= r4951661;
        double r4951663 = r4951601 * r4951601;
        double r4951664 = r4951597 / r4951663;
        double r4951665 = r4951664 * r4951664;
        double r4951666 = r4951604 / r4951665;
        double r4951667 = r4951600 / r4951597;
        double r4951668 = r4951604 * r4951601;
        double r4951669 = r4951668 * r4951668;
        double r4951670 = r4951669 * r4951604;
        double r4951671 = r4951670 / r4951597;
        double r4951672 = r4951667 * r4951671;
        double r4951673 = r4951666 + r4951672;
        double r4951674 = r4951600 / r4951673;
        double r4951675 = 8.815224158947631e+36;
        bool r4951676 = r4951597 <= r4951675;
        double r4951677 = r4951604 * r4951642;
        double r4951678 = r4951677 / r4951597;
        double r4951679 = r4951644 / r4951597;
        double r4951680 = r4951678 * r4951679;
        double r4951681 = r4951680 / r4951648;
        double r4951682 = r4951604 * r4951604;
        double r4951683 = r4951642 * r4951682;
        double r4951684 = r4951604 * r4951683;
        double r4951685 = r4951684 / r4951648;
        double r4951686 = r4951603 / r4951597;
        double r4951687 = r4951686 * r4951686;
        double r4951688 = r4951685 * r4951687;
        double r4951689 = r4951688 * r4951600;
        double r4951690 = r4951681 + r4951689;
        double r4951691 = r4951639 * r4951690;
        double r4951692 = r4951600 / r4951691;
        double r4951693 = cbrt(r4951605);
        double r4951694 = pow(r4951693, r4951606);
        double r4951695 = r4951694 / r4951615;
        double r4951696 = r4951603 * r4951695;
        double r4951697 = cbrt(r4951610);
        double r4951698 = pow(r4951697, r4951606);
        double r4951699 = r4951698 / r4951621;
        double r4951700 = r4951696 * r4951699;
        double r4951701 = r4951613 * r4951700;
        double r4951702 = r4951701 * r4951602;
        double r4951703 = r4951702 * r4951630;
        double r4951704 = r4951597 / r4951613;
        double r4951705 = r4951703 / r4951704;
        double r4951706 = r4951600 / r4951705;
        double r4951707 = r4951676 ? r4951692 : r4951706;
        double r4951708 = r4951662 ? r4951674 : r4951707;
        double r4951709 = r4951634 ? r4951660 : r4951708;
        double r4951710 = r4951599 ? r4951632 : r4951709;
        return r4951710;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if l < -2.385068763139975e+91

   1. Initial program 54.0

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

   3. Applied add-cube-cbrt 54.0

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

   4. Applied unpow-prod-down 54.0

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

   5. Applied times-frac 39.4

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

   6. Applied associate-*l* 39.4

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

   8. Applied sqr-pow 39.4

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

   9. Applied associate-/l* 25.2

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

   11. Applied sqr-pow 25.2

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

   12. Applied add-cube-cbrt 25.2

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

   13. Applied times-frac 25.2

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

   14. Applied add-sqr-sqrt 25.2

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

   15. Applied times-frac 25.2

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

   16. Applied associate-*l* 23.7

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

   if -2.385068763139975e+91 < l < -1.3059910812328375e-155

   1. Initial program 24.9

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

   3. Applied add-cube-cbrt 25.2

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

   4. Applied unpow-prod-down 25.2

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

   5. Applied times-frac 23.3

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

   6. Applied associate-*l* 21.3

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

   8. Applied sqr-pow 21.3

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

   9. Applied associate-/l* 20.1

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

   11. Applied associate-*l/ 19.4

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

   12. Applied associate-*l/ 17.6

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

   13. Applied associate-*l/ 16.1

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

   14. Taylor expanded around -inf 16.2

       \[\leadsto \frac{2}{\color{blue}{\frac{{\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot \left(t \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right)\right)}{{\ell}^{2} \cdot \cos k} \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1} + 2 \cdot \left(\frac{{t}^{3} \cdot \left({\left(\sqrt[3]{-1}\right)}^{9} \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k} \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1}\right)}}\]

   15. Simplified 8.3

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(\frac{\left(t \cdot \left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)\right)\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\cos k \cdot \left(\ell \cdot \ell\right)} + 2 \cdot \frac{\frac{\left(\left(\left(t \cdot \sin k\right) \cdot \left(t \cdot \sin k\right)\right) \cdot t\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\ell \cdot \ell}}{\cos k}\right)}}\]

   if -1.3059910812328375e-155 < l < 7.061293343570338e-144

   1. Initial program 23.7

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

   3. Applied add-cube-cbrt 23.7

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

   4. Applied unpow-prod-down 23.7

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

   5. Applied times-frac 17.2

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

   6. Applied associate-*l* 14.6

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

   8. Applied sqr-pow 14.6

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

   9. Applied associate-/l* 8.8

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

   11. Applied associate-*l/ 8.8

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

   12. Applied associate-*l/ 9.2

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

   13. Applied associate-*l/ 8.6

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

   14. Taylor expanded around 0 33.3

       \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {k}^{4}}{{\ell}^{2}} + 2 \cdot \frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}}\]

   15. Simplified 9.0

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

   if 7.061293343570338e-144 < l < 8.815224158947631e+36

   1. Initial program 22.7

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

   3. Applied add-cube-cbrt 23.0

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

   4. Applied unpow-prod-down 23.0

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

   5. Applied times-frac 21.1

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

   6. Applied associate-*l* 18.1

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

   7. Taylor expanded around -inf 14.1

      \[\leadsto \frac{2}{\color{blue}{\frac{{\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot \left(t \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right)\right)}{{\ell}^{2} \cdot \cos k} \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1} + 2 \cdot \left(\frac{{t}^{3} \cdot \left({\left(\sqrt[3]{-1}\right)}^{9} \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k} \cdot {\left(\frac{1}{{-1}^{3}}\right)}^{1}\right)}}\]

   8. Simplified 12.2

      \[\leadsto \frac{2}{\color{blue}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(\frac{\frac{\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)}{\ell} \cdot \frac{t \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\ell}}{\cos k} + 2 \cdot \left(\frac{t \cdot \left(\left(t \cdot t\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right)}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right)\right)}}\]

   if 8.815224158947631e+36 < l

   1. Initial program 48.6

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

   3. Applied add-cube-cbrt 48.7

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

   4. Applied unpow-prod-down 48.7

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

   5. Applied times-frac 36.3

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

   6. Applied associate-*l* 36.2

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

   8. Applied sqr-pow 36.2

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

   9. Applied associate-/l* 26.4

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

   11. Applied associate-*l/ 24.3

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

   12. Applied associate-*l/ 23.0

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

   13. Applied associate-*l/ 21.3

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

   15. Applied add-cube-cbrt 21.3

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

   16. Applied add-cube-cbrt 21.4

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

   17. Applied cbrt-prod 21.4

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

   18. Applied unpow-prod-down 21.4

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

   19. Applied times-frac 21.4

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

   20. Applied associate-*l* 21.4

       \[\leadsto \frac{2}{\frac{\left(\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \sin k\right)\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.385068763139975120500013847597821926732 \cdot 10^{91}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\frac{\sqrt[3]{\ell}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\right) \cdot \frac{\sqrt{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{\frac{3}{2}}{2}\right)}}}\right)\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\\ \mathbf{elif}\;\ell \le -1.305991081232837483759897853819755248256 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(\frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left(t \cdot \left(\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)\right)\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{\frac{\left(t \cdot \left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right)\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\ell \cdot \ell}}{\cos k}\right)}\\ \mathbf{elif}\;\ell \le 7.061293343570338284594492301637048497777 \cdot 10^{-144}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot k}} + \frac{2}{\ell} \cdot \frac{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right) \cdot t}{\ell}}\\ \mathbf{elif}\;\ell \le 8815224158947631239444986233777815552:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(\frac{\frac{t \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\ell} \cdot \frac{\left(\sin k \cdot k\right) \cdot \left(\sin k \cdot k\right)}{\ell}}{\cos k} + \left(\frac{t \cdot \left({\left(\sqrt[3]{-1}\right)}^{9} \cdot \left(t \cdot t\right)\right)}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right) \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{3}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)\right) \cdot \tan k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))