Derivation

Split input into 3 regimes
if k < 7.823828054360163e-51
1. Initial program 49.4
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Initial simplification33.6
  \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
3. Using strategy rm
4. Applied associate-*l/35.7
  \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\color{blue}{\frac{\ell \cdot \frac{\ell}{t}}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
5. Applied associate-/r/35.7
  \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}} \cdot t}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
6. Applied *-un-lft-identity35.7
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{2}{\tan k}}}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}} \cdot t}}{\frac{k}{t} \cdot \frac{k}{t}}\]
7. Applied times-frac35.6
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}}} \cdot \frac{\frac{2}{\tan k}}{t}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
8. Applied times-frac25.5
  \[\leadsto \color{blue}{\frac{\frac{1}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\frac{2}{\tan k}}{t}}{\frac{k}{t}}}\]
9. Simplified16.5
  \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{\frac{k}{t}}{\frac{\ell}{t}}}} \cdot \frac{\frac{\frac{2}{\tan k}}{t}}{\frac{k}{t}}\]
10. Simplified15.3
  \[\leadsto \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{\frac{k}{t}}{\frac{\ell}{t}}} \cdot \color{blue}{\frac{\frac{2}{k}}{\tan k}}\]
11. Using strategy rm
12. Applied frac-times12.7
  \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{2}{k}}{\frac{\frac{k}{t}}{\frac{\ell}{t}} \cdot \tan k}}\]
13. Simplified8.8
  \[\leadsto \frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{2}{k}}{\color{blue}{\frac{\tan k}{\frac{\ell}{k}}}}\]
14. Using strategy rm
15. Applied *-un-lft-identity8.8
  \[\leadsto \frac{\frac{\frac{\ell}{t}}{\color{blue}{1 \cdot \sin k}} \cdot \frac{2}{k}}{\frac{\tan k}{\frac{\ell}{k}}}\]
16. Applied div-inv8.8
  \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \frac{1}{t}}}{1 \cdot \sin k} \cdot \frac{2}{k}}{\frac{\tan k}{\frac{\ell}{k}}}\]
17. Applied times-frac5.8
  \[\leadsto \frac{\color{blue}{\left(\frac{\ell}{1} \cdot \frac{\frac{1}{t}}{\sin k}\right)} \cdot \frac{2}{k}}{\frac{\tan k}{\frac{\ell}{k}}}\]
18. Applied associate-*l*6.3
  \[\leadsto \frac{\color{blue}{\frac{\ell}{1} \cdot \left(\frac{\frac{1}{t}}{\sin k} \cdot \frac{2}{k}\right)}}{\frac{\tan k}{\frac{\ell}{k}}}\]
19. Simplified6.3
  \[\leadsto \frac{\color{blue}{\ell} \cdot \left(\frac{\frac{1}{t}}{\sin k} \cdot \frac{2}{k}\right)}{\frac{\tan k}{\frac{\ell}{k}}}\]
20. Using strategy rm
21. Applied associate-*r/6.3
  \[\leadsto \frac{\ell \cdot \color{blue}{\frac{\frac{\frac{1}{t}}{\sin k} \cdot 2}{k}}}{\frac{\tan k}{\frac{\ell}{k}}}\]
if 7.823828054360163e-51 < k < 3.749710248922628e+164
1. Initial program 49.1
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Initial simplification28.9
  \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
3. Using strategy rm
4. Applied associate-*l/30.2
  \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\color{blue}{\frac{\ell \cdot \frac{\ell}{t}}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
5. Applied associate-/r/30.2
  \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}} \cdot t}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
6. Applied *-un-lft-identity30.2
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{2}{\tan k}}}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}} \cdot t}}{\frac{k}{t} \cdot \frac{k}{t}}\]
7. Applied times-frac30.0
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}}} \cdot \frac{\frac{2}{\tan k}}{t}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
8. Applied times-frac15.1
  \[\leadsto \color{blue}{\frac{\frac{1}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\frac{2}{\tan k}}{t}}{\frac{k}{t}}}\]
9. Simplified7.4
  \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{\frac{k}{t}}{\frac{\ell}{t}}}} \cdot \frac{\frac{\frac{2}{\tan k}}{t}}{\frac{k}{t}}\]
10. Simplified7.0
  \[\leadsto \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{\frac{k}{t}}{\frac{\ell}{t}}} \cdot \color{blue}{\frac{\frac{2}{k}}{\tan k}}\]
11. Using strategy rm
12. Applied frac-times4.8
  \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{2}{k}}{\frac{\frac{k}{t}}{\frac{\ell}{t}} \cdot \tan k}}\]
13. Simplified1.9
  \[\leadsto \frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{2}{k}}{\color{blue}{\frac{\tan k}{\frac{\ell}{k}}}}\]
14. Using strategy rm
15. Applied frac-times1.9
  \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t} \cdot 2}{\sin k \cdot k}}}{\frac{\tan k}{\frac{\ell}{k}}}\]
if 3.749710248922628e+164 < k
1. Initial program 38.1
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Initial simplification21.8
  \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
3. Using strategy rm
4. Applied associate-*l/23.2
  \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\color{blue}{\frac{\ell \cdot \frac{\ell}{t}}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
5. Applied associate-/r/23.2
  \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}} \cdot t}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
6. Applied *-un-lft-identity23.2
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{2}{\tan k}}}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}} \cdot t}}{\frac{k}{t} \cdot \frac{k}{t}}\]
7. Applied times-frac23.2
  \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}}} \cdot \frac{\frac{2}{\tan k}}{t}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
8. Applied times-frac18.5
  \[\leadsto \color{blue}{\frac{\frac{1}{\frac{\sin k \cdot t}{\ell \cdot \frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\frac{2}{\tan k}}{t}}{\frac{k}{t}}}\]
9. Simplified12.3
  \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{\frac{k}{t}}{\frac{\ell}{t}}}} \cdot \frac{\frac{\frac{2}{\tan k}}{t}}{\frac{k}{t}}\]
10. Simplified12.3
  \[\leadsto \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{\frac{k}{t}}{\frac{\ell}{t}}} \cdot \color{blue}{\frac{\frac{2}{k}}{\tan k}}\]
11. Using strategy rm
12. Applied frac-times11.9
  \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{2}{k}}{\frac{\frac{k}{t}}{\frac{\ell}{t}} \cdot \tan k}}\]
13. Simplified6.9
  \[\leadsto \frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{2}{k}}{\color{blue}{\frac{\tan k}{\frac{\ell}{k}}}}\]
14. Taylor expanded around -inf 5.0
  \[\leadsto \frac{\color{blue}{2 \cdot \frac{\ell}{t \cdot \left(\sin k \cdot k\right)}}}{\frac{\tan k}{\frac{\ell}{k}}}\]
Recombined 3 regimes into one program.
Final simplification5.0
\[\leadsto \begin{array}{l} \mathbf{if}\;k \le 7.823828054360163 \cdot 10^{-51}:\\ \;\;\;\;\frac{\ell \cdot \frac{2 \cdot \frac{\frac{1}{t}}{\sin k}}{k}}{\frac{\tan k}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \le 3.749710248922628 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot 2}{\sin k \cdot k}}{\frac{\tan k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \left(\sin k \cdot k\right)} \cdot 2}{\frac{\tan k}{\frac{\ell}{k}}}\\ \end{array}\]

Error

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Derivation

`if k < 7.823828054360163e-51`

`if 7.823828054360163e-51 < k < 3.749710248922628e+164`

`if 3.749710248922628e+164 < k`

Runtime

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