Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 9.9

Time: 7.9m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -1.483471567911308015546022244990908340338 \cdot 10^{137}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{\left(\sqrt[3]{k}\right)}^{2}}\right)}^{1} \cdot \left(\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right)\right)\right)\\ \mathbf{elif}\;k \le -9.730288504549286413960983657697353219034 \cdot 10^{-144}:\\ \;\;\;\;\left(\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right) \cdot \left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right) \cdot 2\\ \mathbf{elif}\;k \le 6.303690353095152979093645195617976968905 \cdot 10^{-135}:\\ \;\;\;\;\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;k \le 1.488110455716832872059851006586872843391 \cdot 10^{172}:\\ \;\;\;\;\left(\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right) \cdot \left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r12786739 = 2.0;
        double r12786740 = t;
        double r12786741 = 3.0;
        double r12786742 = pow(r12786740, r12786741);
        double r12786743 = l;
        double r12786744 = r12786743 * r12786743;
        double r12786745 = r12786742 / r12786744;
        double r12786746 = k;
        double r12786747 = sin(r12786746);
        double r12786748 = r12786745 * r12786747;
        double r12786749 = tan(r12786746);
        double r12786750 = r12786748 * r12786749;
        double r12786751 = 1.0;
        double r12786752 = r12786746 / r12786740;
        double r12786753 = pow(r12786752, r12786739);
        double r12786754 = r12786751 + r12786753;
        double r12786755 = r12786754 - r12786751;
        double r12786756 = r12786750 * r12786755;
        double r12786757 = r12786739 / r12786756;
        return r12786757;
}

↓

double f(double t, double l, double k) {
        double r12786758 = k;
        double r12786759 = -1.483471567911308e+137;
        bool r12786760 = r12786758 <= r12786759;
        double r12786761 = 2.0;
        double r12786762 = 1.0;
        double r12786763 = cbrt(r12786758);
        double r12786764 = r12786763 * r12786763;
        double r12786765 = pow(r12786764, r12786761);
        double r12786766 = r12786762 / r12786765;
        double r12786767 = 1.0;
        double r12786768 = pow(r12786766, r12786767);
        double r12786769 = pow(r12786763, r12786761);
        double r12786770 = r12786762 / r12786769;
        double r12786771 = pow(r12786770, r12786767);
        double r12786772 = cos(r12786758);
        double r12786773 = cbrt(r12786772);
        double r12786774 = r12786773 * r12786773;
        double r12786775 = sin(r12786758);
        double r12786776 = l;
        double r12786777 = r12786775 / r12786776;
        double r12786778 = r12786774 / r12786777;
        double r12786779 = t;
        double r12786780 = pow(r12786779, r12786767);
        double r12786781 = r12786762 / r12786780;
        double r12786782 = pow(r12786781, r12786767);
        double r12786783 = r12786773 / r12786777;
        double r12786784 = r12786782 * r12786783;
        double r12786785 = r12786778 * r12786784;
        double r12786786 = r12786771 * r12786785;
        double r12786787 = r12786768 * r12786786;
        double r12786788 = r12786761 * r12786787;
        double r12786789 = -9.730288504549286e-144;
        bool r12786790 = r12786758 <= r12786789;
        double r12786791 = pow(r12786758, r12786761);
        double r12786792 = r12786762 / r12786791;
        double r12786793 = pow(r12786792, r12786767);
        double r12786794 = r12786793 * r12786778;
        double r12786795 = r12786784 * r12786794;
        double r12786796 = r12786795 * r12786761;
        double r12786797 = 6.303690353095153e-135;
        bool r12786798 = r12786758 <= r12786797;
        double r12786799 = 2.0;
        double r12786800 = r12786761 / r12786799;
        double r12786801 = pow(r12786758, r12786800);
        double r12786802 = r12786762 / r12786801;
        double r12786803 = pow(r12786802, r12786767);
        double r12786804 = r12786803 * r12786785;
        double r12786805 = r12786804 * r12786803;
        double r12786806 = r12786805 * r12786761;
        double r12786807 = 1.4881104557168329e+172;
        bool r12786808 = r12786758 <= r12786807;
        double r12786809 = r12786808 ? r12786796 : r12786806;
        double r12786810 = r12786798 ? r12786806 : r12786809;
        double r12786811 = r12786790 ? r12786796 : r12786810;
        double r12786812 = r12786760 ? r12786788 : r12786811;
        return r12786812;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -1.483471567911308e+137

   1. Initial program 40.6

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

   2. Simplified 34.1

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

   3. Taylor expanded around inf 23.5

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

   5. Applied *-un-lft-identity 23.5

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

   6. Applied times-frac 23.5

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

   7. Applied unpow-prod-down 23.5

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

   8. Applied associate-*l* 23.7

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

   9. Simplified 23.7

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

   11. Applied add-cube-cbrt 23.8

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

   12. Applied times-frac 23.8

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

   13. Applied associate-*l* 22.1

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

   15. Applied add-cube-cbrt 22.1

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

   16. Applied unpow-prod-down 22.1

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

   17. Applied *-un-lft-identity 22.1

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

   18. Applied times-frac 21.8

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

   19. Applied unpow-prod-down 21.8

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

   20. Applied associate-*l* 16.9

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

   if -1.483471567911308e+137 < k < -9.730288504549286e-144 or 6.303690353095153e-135 < k < 1.4881104557168329e+172

   1. Initial program 53.4

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

   2. Simplified 43.6

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

   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 *-un-lft-identity 18.4

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

   6. Applied times-frac 18.3

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

   7. Applied unpow-prod-down 18.3

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

   8. Applied associate-*l* 18.4

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

   9. Simplified 15.9

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

   11. Applied add-cube-cbrt 16.0

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

   12. Applied times-frac 15.8

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

   13. Applied associate-*l* 9.0

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

   15. Applied associate-*r* 4.4

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

   if -9.730288504549286e-144 < k < 6.303690353095153e-135 or 1.4881104557168329e+172 < k

   1. Initial program 44.3

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

   2. Simplified 40.5

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

   3. Taylor expanded around inf 31.0

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

   5. Applied *-un-lft-identity 31.0

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

   6. Applied times-frac 31.2

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

   7. Applied unpow-prod-down 31.2

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

   8. Applied associate-*l* 31.2

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

   9. Simplified 30.1

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

   11. Applied add-cube-cbrt 30.1

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

   12. Applied times-frac 30.0

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

   13. Applied associate-*l* 28.9

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

   15. Applied sqr-pow 28.9

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

   16. Applied *-un-lft-identity 28.9

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

   17. Applied times-frac 28.9

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

   18. Applied unpow-prod-down 28.9

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

   19. Applied associate-*l* 16.7

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.483471567911308015546022244990908340338 \cdot 10^{137}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{\left(\sqrt[3]{k}\right)}^{2}}\right)}^{1} \cdot \left(\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right)\right)\right)\\ \mathbf{elif}\;k \le -9.730288504549286413960983657697353219034 \cdot 10^{-144}:\\ \;\;\;\;\left(\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right) \cdot \left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right) \cdot 2\\ \mathbf{elif}\;k \le 6.303690353095152979093645195617976968905 \cdot 10^{-135}:\\ \;\;\;\;\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;k \le 1.488110455716832872059851006586872843391 \cdot 10^{172}:\\ \;\;\;\;\left(\left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right) \cdot \left({\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sin k}{\ell}} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\sin k}{\ell}}\right)\right)\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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))))