Derivation

Split input into 2 regimes
if k < -1.449716848200777e-87 or 2.9444387605578513e-145 < k
1. Initial program 46.3
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Taylor expanded around -inf 62.8
  \[\leadsto \frac{2}{\left(\color{blue}{\frac{e^{3 \cdot \left(\log -1 - \log \left(\frac{-1}{t}\right)\right)} \cdot \sin k}{{\ell}^{2}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
3. Applied simplify36.2
  \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \frac{\sin k}{\ell}}}\]
4. Taylor expanded around inf 19.8
  \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}}\]
5. Taylor expanded around -inf 62.8
  \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{k}\right)\right)} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}}\]
6. Applied simplify7.3
  \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}}\]
7. Using strategy rm
8. Applied associate-/l*1.5
  \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{\frac{t}{\frac{\ell}{k}}}} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}\]
if -1.449716848200777e-87 < k < 2.9444387605578513e-145
1. Initial program 62.3
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Taylor expanded around -inf 62.4
  \[\leadsto \frac{2}{\left(\color{blue}{\frac{e^{3 \cdot \left(\log -1 - \log \left(\frac{-1}{t}\right)\right)} \cdot \sin k}{{\ell}^{2}}} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
3. Applied simplify59.0
  \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \tan k\right) \cdot \frac{\sin k}{\ell}}}\]
4. Taylor expanded around inf 54.7
  \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}}\]
5. Taylor expanded around -inf 62.4
  \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{e^{2 \cdot \left(\log -1 - \log \left(\frac{-1}{k}\right)\right)} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}}\]
6. Applied simplify37.6
  \[\leadsto \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}}\]
7. Using strategy rm
8. Applied div-inv37.6
  \[\leadsto \color{blue}{\left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t}\right)} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}\]
9. Applied associate-*l*39.4
  \[\leadsto \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t} \cdot \frac{\cos k + \cos k}{\sin k \cdot \sin k}\right)}\]
10. Applied simplify17.6
  \[\leadsto \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\left(\frac{\cos k}{\sin k} \cdot \frac{\frac{2}{t}}{\sin k}\right)}\]
Recombined 2 regimes into one program.

Error

Derivation

`if k < -1.449716848200777e-87 or 2.9444387605578513e-145 < k`

`if -1.449716848200777e-87 < k < 2.9444387605578513e-145`

Runtime