Average Error: 48.6 → 10.8
Time: 1.6m
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 -1.2834311638414869 \cdot 10^{152}:\\ \;\;\;\;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}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le -2.02559849820690049 \cdot 10^{-183}:\\ \;\;\;\;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 2.3681957710515152 \cdot 10^{-158}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \le 5.359847357268255 \cdot 10^{152}:\\ \;\;\;\;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{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt{\ell}}}{\sqrt{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\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 -1.2834311638414869 \cdot 10^{152}:\\
\;\;\;\;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}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\

\mathbf{elif}\;\ell \le -2.02559849820690049 \cdot 10^{-183}:\\
\;\;\;\;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 2.3681957710515152 \cdot 10^{-158}:\\
\;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\

\mathbf{elif}\;\ell \le 5.359847357268255 \cdot 10^{152}:\\
\;\;\;\;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{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\

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

\end{array}
double code(double t, double l, double k) {
	return (2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0)));
}
↓
double code(double t, double l, double k) {
	double VAR;
	if ((l <= -1.283431163841487e+152)) {
		VAR = (2.0 * ((pow((1.0 / (pow(k, (2.0 / 2.0)) * (pow(k, (2.0 / 2.0)) * pow(t, 1.0)))), 1.0) * ((1.0 / (pow(cbrt(sin(k)), 4.0) / l)) / 1.0)) * ((cos(k) / (1.0 / l)) / pow(cbrt(sin(k)), 2.0))));
	} else {
		double VAR_1;
		if ((l <= -2.0255984982069005e-183)) {
			VAR_1 = (2.0 * (pow((1.0 / pow(k, (2.0 / 2.0))), 1.0) * (pow((1.0 / (pow(k, (2.0 / 2.0)) * pow(t, 1.0))), 1.0) * ((cos(k) / ((pow(cbrt(sin(k)), 4.0) / l) / l)) / pow(cbrt(sin(k)), 2.0)))));
		} else {
			double VAR_2;
			if ((l <= 2.368195771051515e-158)) {
				VAR_2 = (2.0 * ((pow((1.0 / (pow(k, (2.0 / 2.0)) * (pow(k, (2.0 / 2.0)) * pow(t, 1.0)))), 1.0) * (((cbrt(cos(k)) * cbrt(cos(k))) / ((cbrt((pow(cbrt(sin(k)), 4.0) / l)) * cbrt((pow(cbrt(sin(k)), 4.0) / l))) / (cbrt(l) * cbrt(l)))) / (cbrt(pow(cbrt(sin(k)), 2.0)) * cbrt(pow(cbrt(sin(k)), 2.0))))) * ((cbrt(cos(k)) / (cbrt((pow(cbrt(sin(k)), 4.0) / l)) / cbrt(l))) / cbrt(pow(cbrt(sin(k)), 2.0)))));
			} else {
				double VAR_3;
				if ((l <= 5.359847357268255e+152)) {
					VAR_3 = (2.0 * (pow((1.0 / pow(k, (2.0 / 2.0))), 1.0) * (pow((1.0 / (pow(k, (2.0 / 2.0)) * pow(t, 1.0))), 1.0) * ((cos(k) * pow(l, 2.0)) / pow(sin(k), 2.0)))));
				} else {
					VAR_3 = (2.0 * ((pow((1.0 / (pow(k, (2.0 / 2.0)) * (pow(k, (2.0 / 2.0)) * pow(t, 1.0)))), 1.0) * ((1.0 / ((1.0 / sqrt(l)) / sqrt(l))) / 1.0)) * ((cos(k) / ((pow(cbrt(sin(k)), 4.0) / sqrt(l)) / sqrt(l))) / pow(cbrt(sin(k)), 2.0))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 5 regimes

  2. if l < -1.283431163841487e+152

    1. Initial program 63.8

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

    2. Simplified63.6

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

    3. Taylor expanded around inf 63.5

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

    5. Applied sqr-pow63.5

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

    6. Applied associate-*l*63.5

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

    8. Applied add-cube-cbrt63.5

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

    9. Applied unpow-prod-down63.5

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

    10. Applied associate-/r*63.5

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

    11. Simplified63.5

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

    13. Applied *-un-lft-identity63.5

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

    14. Applied div-inv63.5

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

    15. Applied *-un-lft-identity63.5

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

    16. Applied times-frac63.5

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

    17. Applied times-frac63.5

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

    18. Applied associate-*r*40.6

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

    if -1.283431163841487e+152 < l < -2.0255984982069005e-183

    1. Initial program 46.0

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

    2. Simplified35.8

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

    3. Taylor expanded around inf 11.8

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

    5. Applied sqr-pow11.8

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

    6. Applied associate-*l*8.0

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

    8. Applied add-cube-cbrt8.5

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

    9. Applied unpow-prod-down8.5

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

    10. Applied associate-/r*8.1

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

    11. Simplified7.3

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

    13. Applied *-un-lft-identity7.3

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

    14. Applied times-frac7.0

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

    15. Applied unpow-prod-down7.0

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

    16. Applied associate-*l*3.9

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

    if -2.0255984982069005e-183 < l < 2.368195771051515e-158

    1. Initial program 46.5

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

    2. Simplified38.4

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

    3. Taylor expanded around inf 20.2

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

    5. Applied sqr-pow20.2

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

    6. Applied associate-*l*20.2

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

    8. Applied add-cube-cbrt20.2

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

    9. Applied unpow-prod-down20.2

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

    10. Applied associate-/r*20.0

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

    11. Simplified14.2

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

    13. Applied add-cube-cbrt14.2

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

    14. Applied add-cube-cbrt14.2

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

    15. Applied add-cube-cbrt14.2

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

    16. Applied times-frac14.2

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

    17. Applied add-cube-cbrt14.2

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

    18. Applied times-frac13.7

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

    19. Applied times-frac11.4

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

    20. Applied associate-*r*8.0

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

    if 2.368195771051515e-158 < l < 5.359847357268255e+152

    1. Initial program 44.7

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

    2. Simplified34.3

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

    3. Taylor expanded around inf 10.7

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

    5. Applied sqr-pow10.7

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

    6. Applied associate-*l*6.7

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

    8. Applied *-un-lft-identity6.7

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

    9. Applied times-frac6.3

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

    10. Applied unpow-prod-down6.3

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

    11. Applied associate-*l*3.5

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

    if 5.359847357268255e+152 < l

    1. Initial program 63.8

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

    2. Simplified63.8

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

    3. Taylor expanded around inf 63.7

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

    5. Applied sqr-pow63.7

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

    6. Applied associate-*l*63.7

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

    8. Applied add-cube-cbrt63.7

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

    9. Applied unpow-prod-down63.7

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

    10. Applied associate-/r*63.7

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

    11. Simplified63.7

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

    13. Applied *-un-lft-identity63.7

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

    14. Applied add-sqr-sqrt63.7

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

    15. Applied add-sqr-sqrt63.7

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

    16. Applied *-un-lft-identity63.7

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

    17. Applied times-frac63.7

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

    18. Applied times-frac63.7

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

    19. Applied *-un-lft-identity63.7

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

    20. Applied times-frac63.7

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

    21. Applied times-frac63.7

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

    22. Applied associate-*r*40.2

      \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt{\ell}}}{\sqrt{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.2834311638414869 \cdot 10^{152}:\\ \;\;\;\;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}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le -2.02559849820690049 \cdot 10^{-183}:\\ \;\;\;\;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 2.3681957710515152 \cdot 10^{-158}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\sqrt[3]{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{\sqrt[3]{\ell}}}}{\sqrt[3]{{\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\\ \mathbf{elif}\;\ell \le 5.359847357268255 \cdot 10^{152}:\\ \;\;\;\;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{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{1}{\frac{\frac{1}{\sqrt{\ell}}}{\sqrt{\ell}}}}{1}\right) \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\sqrt{\ell}}}{\sqrt{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))