Derivation

Split input into 2 regimes
if t < -9.547084835904648e+35 or 1.8225529679406173e+56 < t
1. Initial program 23.2
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
2. Initial simplification7.4
  \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*}\]
3. Using strategy rm
4. Applied times-frac7.6
  \[\leadsto \frac{2}{\color{blue}{\left(\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}\right)} \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*}\]
5. Applied associate-*l*7.2
  \[\leadsto \frac{2}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*\right)}}\]
6. Using strategy rm
7. Applied associate-*l/2.4
  \[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \color{blue}{\frac{t \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*}{\frac{\ell}{t}}}}\]
8. Applied associate-*r/2.8
  \[\leadsto \frac{2}{\color{blue}{\frac{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(t \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*\right)}{\frac{\ell}{t}}}}\]
9. Applied associate-/r/2.6
  \[\leadsto \color{blue}{\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(t \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*\right)} \cdot \frac{\ell}{t}}\]
if -9.547084835904648e+35 < t < 1.8225529679406173e+56
1. Initial program 42.5
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
2. Initial simplification31.6
  \[\leadsto \frac{2}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*}\]
3. Using strategy rm
4. Applied times-frac28.8
  \[\leadsto \frac{2}{\color{blue}{\left(\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}\right)} \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*}\]
5. Applied associate-*l*25.0
  \[\leadsto \frac{2}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left(\frac{t}{\frac{\ell}{t}} \cdot (\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_*\right)}}\]
6. Taylor expanded around -inf 16.4
  \[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \color{blue}{\left(2 \cdot \frac{{t}^{2} \cdot \sin k}{\ell \cdot \cos k} + \frac{\sin k \cdot {k}^{2}}{\cos k \cdot \ell}\right)}}\]
7. Simplified12.0
  \[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \color{blue}{\left((2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(k \cdot \frac{k}{\ell}\right))_* \cdot \frac{\sin k}{\cos k}\right)}}\]
8. Using strategy rm
9. Applied associate-*r/12.0
  \[\leadsto \frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \color{blue}{\frac{(2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(k \cdot \frac{k}{\ell}\right))_* \cdot \sin k}{\cos k}}}\]
10. Applied associate-*r/12.0
  \[\leadsto \frac{2}{\color{blue}{\frac{\frac{\sin k}{\frac{\ell}{t}} \cdot \left((2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(k \cdot \frac{k}{\ell}\right))_* \cdot \sin k\right)}{\cos k}}}\]
11. Applied associate-/r/12.0
  \[\leadsto \color{blue}{\frac{2}{\frac{\sin k}{\frac{\ell}{t}} \cdot \left((2 \cdot \left(\frac{t}{\ell} \cdot t\right) + \left(k \cdot \frac{k}{\ell}\right))_* \cdot \sin k\right)} \cdot \cos k}\]
12. Taylor expanded around inf 31.7
  \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{{\ell}^{2}} + \frac{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}{{\ell}^{2}}}} \cdot \cos k\]
13. Simplified5.0
  \[\leadsto \frac{2}{\color{blue}{\left(\frac{\sin k}{\ell} \cdot \sin k\right) \cdot \left(\frac{t \cdot k}{\frac{\ell}{k}} + \frac{{t}^{3}}{\frac{\ell}{2}}\right)}} \cdot \cos k\]
Recombined 2 regimes into one program.
Final simplification3.7
\[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.547084835904648 \cdot 10^{+35} \lor \neg \left(t \le 1.8225529679406173 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left(\tan k\right) + \left(\tan k\right))_* \cdot t\right) \cdot \frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k \cdot t}{\frac{\ell}{k}} + \frac{{t}^{3}}{\frac{\ell}{2}}\right) \cdot \left(\sin k \cdot \frac{\sin k}{\ell}\right)} \cdot \cos k\\ \end{array}\]

Error

Derivation

`if t < -9.547084835904648e+35 or 1.8225529679406173e+56 < t`

`if -9.547084835904648e+35 < t < 1.8225529679406173e+56`

Runtime