Derivation

Split input into 2 regimes
if k < -1.5179674201912627e-32 or 3.3605956880041184e-65 < k
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. Initial simplification26.5
  \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}}\]
3. Using strategy rm
4. Applied times-frac26.4
  \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
5. Applied times-frac15.6
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]
6. Simplified15.5
  \[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
7. Using strategy rm
8. Applied *-un-lft-identity15.5
  \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]
9. Applied *-un-lft-identity15.5
  \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{1 \cdot \frac{k}{t}}\]
10. Applied times-frac15.4
  \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{1 \cdot \frac{k}{t}}\]
11. Applied times-frac9.8
  \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{1} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\right)}\]
12. Applied associate-*r*8.2
  \[\leadsto \color{blue}{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]
13. Simplified4.0
  \[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\tan k}}\]
14. Using strategy rm
15. Applied frac-times4.0
  \[\leadsto \color{blue}{\frac{\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{1}}{\sin k \cdot 1}} \cdot \frac{\frac{\ell}{k}}{\tan k}\]
16. Applied associate-*l/4.1
  \[\leadsto \color{blue}{\frac{\left(\frac{2}{k} \cdot \frac{\frac{\ell}{t}}{1}\right) \cdot \frac{\frac{\ell}{k}}{\tan k}}{\sin k \cdot 1}}\]
17. Simplified0.5
  \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{\frac{\tan k}{\frac{\ell}{k}}}}}{\sin k \cdot 1}\]
18. Using strategy rm
19. Applied associate-/r/0.5
  \[\leadsto \frac{\frac{\frac{\ell}{k} \cdot \frac{2}{t}}{\color{blue}{\frac{\tan k}{\ell} \cdot k}}}{\sin k \cdot 1}\]
if -1.5179674201912627e-32 < k < 3.3605956880041184e-65
1. Initial program 61.4
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Initial simplification51.0
  \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t}}\]
3. Using strategy rm
4. Applied times-frac50.9
  \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
5. Applied times-frac44.2
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]
6. Simplified40.6
  \[\leadsto \color{blue}{\frac{\frac{2}{k}}{\sin k}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\]
7. Using strategy rm
8. Applied *-un-lft-identity40.6
  \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{1 \cdot \frac{k}{t}}}\]
9. Applied *-un-lft-identity40.6
  \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{1 \cdot \frac{k}{t}}\]
10. Applied times-frac36.2
  \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{1 \cdot \frac{k}{t}}\]
11. Applied times-frac29.7
  \[\leadsto \frac{\frac{2}{k}}{\sin k} \cdot \color{blue}{\left(\frac{\frac{\frac{\ell}{t}}{1}}{1} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}\right)}\]
12. Applied associate-*r*27.9
  \[\leadsto \color{blue}{\left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\frac{k}{t}}}\]
13. Simplified22.6
  \[\leadsto \left(\frac{\frac{2}{k}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{1}\right) \cdot \color{blue}{\frac{\frac{\ell}{k}}{\tan k}}\]
14. Taylor expanded around 0 19.7
  \[\leadsto \color{blue}{\left(2 \cdot \frac{\ell}{t \cdot {k}^{2}} + \frac{1}{3} \cdot \frac{\ell}{t}\right)} \cdot \frac{\frac{\ell}{k}}{\tan k}\]
15. Simplified0.6
  \[\leadsto \color{blue}{\frac{(\left(\frac{\ell}{k}\right) \cdot \left(\frac{2}{k}\right) + \left(\frac{1}{3} \cdot \ell\right))_*}{t}} \cdot \frac{\frac{\ell}{k}}{\tan k}\]
Recombined 2 regimes into one program.
Final simplification0.5
\[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.5179674201912627 \cdot 10^{-32} \lor \neg \left(k \le 3.3605956880041184 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{\frac{\frac{2}{t} \cdot \frac{\ell}{k}}{k \cdot \frac{\tan k}{\ell}}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{(\left(\frac{\ell}{k}\right) \cdot \left(\frac{2}{k}\right) + \left(\frac{1}{3} \cdot \ell\right))_*}{t}\\ \end{array}\]

Error

Derivation

`if k < -1.5179674201912627e-32 or 3.3605956880041184e-65 < k`

`if -1.5179674201912627e-32 < k < 3.3605956880041184e-65`

Runtime