Derivation

Split input into 2 regimes
if t < 7.547122716476984e+29
1. Initial program 47.1
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied add-sqr-sqrt47.1
  \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}}\]
4. Applied simplify47.1
  \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left|\frac{k}{t}\right|} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}\]
5. Applied simplify41.3
  \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \color{blue}{\left|\frac{k}{t}\right|}\right)}\]
6. Using strategy rm
7. Applied unpow341.3
  \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
8. Applied times-frac35.4
  \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
9. Using strategy rm
10. Applied associate-/l*31.0
  \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
11. Using strategy rm
12. Applied tan-quot31.0
  \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
13. Applied associate-*l/31.0
  \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot \frac{t}{\ell}}{\frac{\ell}{t}}} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
14. Applied associate-*l/30.5
  \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k}{\frac{\ell}{t}}} \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
15. Applied frac-times30.3
  \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}{\frac{\ell}{t} \cdot \cos k}} \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
16. Applied associate-*l/27.9
  \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}{\frac{\ell}{t} \cdot \cos k}}}\]
if 7.547122716476984e+29 < t
1. Initial program 46.1
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied add-sqr-sqrt46.1
  \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}}\]
4. Applied simplify46.1
  \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left|\frac{k}{t}\right|} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}\]
5. Applied simplify32.9
  \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \color{blue}{\left|\frac{k}{t}\right|}\right)}\]
6. Using strategy rm
7. Applied unpow332.9
  \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
8. Applied times-frac28.1
  \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
9. Using strategy rm
10. Applied associate-/l*24.7
  \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
11. Using strategy rm
12. Applied associate-*r*17.4
  \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|}}\]
Recombined 2 regimes into one program.

Error

Derivation

`if t < 7.547122716476984e+29`

`if 7.547122716476984e+29 < t`

Runtime