Derivation

Split input into 2 regimes
if k < -1.4963373848521358e-53 or 3.561642584406255e-78 < k
1. Initial program 30.5
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
2. Initial simplification18.4
  \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
3. Using strategy rm
4. Applied times-frac18.2
  \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
5. Applied associate-/l*16.4
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}}\]
6. Taylor expanded around inf 21.9
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{\sin k \cdot {k}^{2}}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{{\ell}^{2} \cdot \cos k}}}\]
7. Simplified5.1
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{(2 \cdot \left(\frac{\frac{t}{\ell} \cdot \frac{t}{\ell}}{\frac{\cos k}{\sin k}}\right) + \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right))_*}}\]
8. Using strategy rm
9. Applied div-inv5.1
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{(2 \cdot \left(\frac{\frac{t}{\ell} \cdot \frac{t}{\ell}}{\color{blue}{\cos k \cdot \frac{1}{\sin k}}}\right) + \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right))_*}\]
10. Applied times-frac5.1
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{(2 \cdot \color{blue}{\left(\frac{\frac{t}{\ell}}{\cos k} \cdot \frac{\frac{t}{\ell}}{\frac{1}{\sin k}}\right)} + \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right))_*}\]
11. Simplified5.1
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{(2 \cdot \left(\frac{\frac{t}{\ell}}{\cos k} \cdot \color{blue}{\left(\sin k \cdot \frac{t}{\ell}\right)}\right) + \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right))_*}\]
12. Using strategy rm
13. Applied div-inv5.1
  \[\leadsto \frac{\color{blue}{\frac{2}{t} \cdot \frac{1}{\sin k}}}{(2 \cdot \left(\frac{\frac{t}{\ell}}{\cos k} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) + \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right))_*}\]
14. Applied associate-/l*5.1
  \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{(2 \cdot \left(\frac{\frac{t}{\ell}}{\cos k} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) + \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\sin k}{\cos k}\right))_*}{\frac{1}{\sin k}}}}\]
if -1.4963373848521358e-53 < k < 3.561642584406255e-78
1. Initial program 33.5
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
2. Initial simplification41.3
  \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
3. Using strategy rm
4. Applied times-frac17.0
  \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\]
5. Applied associate-/l*16.6
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}}\]
6. Using strategy rm
7. Applied add-cube-cbrt16.9
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}\right) \cdot \sqrt[3]{\tan k}}}}}\]
8. Applied times-frac7.4
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\color{blue}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}}\]
9. Applied *-un-lft-identity7.4
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\color{blue}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}\]
10. Applied times-frac7.2
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}}\]
11. Applied *-un-lft-identity7.2
  \[\leadsto \frac{\frac{\frac{2}{t}}{\color{blue}{1 \cdot \sin k}}}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}\]
12. Applied div-inv7.2
  \[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{1}{t}}}{1 \cdot \sin k}}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}\]
13. Applied times-frac7.2
  \[\leadsto \frac{\color{blue}{\frac{2}{1} \cdot \frac{\frac{1}{t}}{\sin k}}}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}\]
14. Applied times-frac4.6
  \[\leadsto \color{blue}{\frac{\frac{2}{1}}{\frac{1}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}} \cdot \frac{\frac{\frac{1}{t}}{\sin k}}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}}\]
15. Simplified4.6
  \[\leadsto \color{blue}{\frac{\frac{\ell}{t} \cdot 2}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}} \cdot \frac{\frac{\frac{1}{t}}{\sin k}}{\frac{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}{\frac{\frac{\ell}{t}}{\sqrt[3]{\tan k}}}}\]
16. Simplified2.8
  \[\leadsto \frac{\frac{\ell}{t} \cdot 2}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \color{blue}{\frac{\frac{\frac{\frac{\ell}{t}}{\sin k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{t \cdot \sqrt[3]{\tan k}}}\]
Recombined 2 regimes into one program.
Final simplification4.5
\[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.4963373848521358 \cdot 10^{-53} \lor \neg \left(k \le 3.561642584406255 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{(2 \cdot \left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \frac{\frac{t}{\ell}}{\cos k}\right) + \left(\frac{\sin k}{\cos k} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right))_*}{\frac{1}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot 2}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \frac{\frac{\frac{\frac{\ell}{t}}{\sin k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{\sqrt[3]{\tan k} \cdot t}\\ \end{array}\]

Error

Derivation

`if k < -1.4963373848521358e-53 or 3.561642584406255e-78 < k`

`if -1.4963373848521358e-53 < k < 3.561642584406255e-78`

Runtime