Derivation

Split input into 3 regimes
if k < -1.704038783994604e-91
1. Initial program 45.9
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied add-cbrt-cube47.8
  \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
4. Applied simplify34.5
  \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]
5. Taylor expanded around inf 28.1
  \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{3}}}\]
6. Applied simplify7.3
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}}\]
7. Using strategy rm
8. Applied div-inv7.4
  \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{\ell}} \cdot \frac{1}{\frac{k}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
9. Applied times-frac0.7
  \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\frac{1}{\frac{k}{\ell}}}{\sin k \cdot \sin k}}\]
10. Applied simplify0.6
  \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{\sin k \cdot \sin k}}\]
11. Using strategy rm
12. Applied *-un-lft-identity0.6
  \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\color{blue}{1 \cdot \frac{\ell}{k}}}{\sin k \cdot \sin k}\]
13. Applied times-frac0.7
  \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \color{blue}{\left(\frac{1}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}\right)}\]
if -1.704038783994604e-91 < k < 6.533045175321327e-80
1. Initial program 62.1
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied add-cbrt-cube62.3
  \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
4. Applied simplify60.5
  \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]
5. Taylor expanded around inf 55.5
  \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{3}}}\]
6. Applied simplify30.2
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}}\]
7. Using strategy rm
8. Applied div-inv30.2
  \[\leadsto \frac{\frac{\frac{2}{\frac{k}{\ell}}}{\color{blue}{k \cdot \frac{1}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
9. Applied add-cube-cbrt30.5
  \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}\right) \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}}{k \cdot \frac{1}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
10. Applied times-frac31.9
  \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{k} \cdot \frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{1}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
11. Applied times-frac26.3
  \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{k}}{\frac{t}{\cos k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{\frac{1}{\ell}}}{\sin k \cdot \sin k}}\]
12. Applied simplify2.7
  \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\frac{k}{\ell}}} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}}{k}}{\frac{t}{\cos k}} \cdot \color{blue}{\left(\frac{\frac{\ell}{\sin k}}{\sin k} \cdot \sqrt[3]{\frac{2}{\frac{k}{\ell}}}\right)}\]
if 6.533045175321327e-80 < k
1. Initial program 45.6
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied add-cbrt-cube47.1
  \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
4. Applied simplify33.4
  \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*\right) \cdot \frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]
5. Taylor expanded around inf 26.7
  \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k}\right)}}^{3}}}\]
6. Applied simplify7.5
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{k}{\ell}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}}\]
7. Using strategy rm
8. Applied div-inv7.5
  \[\leadsto \frac{\color{blue}{\frac{2}{\frac{k}{\ell}} \cdot \frac{1}{\frac{k}{\ell}}}}{\frac{t}{\cos k} \cdot \left(\sin k \cdot \sin k\right)}\]
9. Applied times-frac0.6
  \[\leadsto \color{blue}{\frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \frac{\frac{1}{\frac{k}{\ell}}}{\sin k \cdot \sin k}}\]
10. Applied simplify0.5
  \[\leadsto \frac{\frac{2}{\frac{k}{\ell}}}{\frac{t}{\cos k}} \cdot \color{blue}{\frac{\frac{\ell}{k}}{\sin k \cdot \sin k}}\]
Recombined 3 regimes into one program.

Error

Derivation

`if k < -1.704038783994604e-91`

`if -1.704038783994604e-91 < k < 6.533045175321327e-80`

`if 6.533045175321327e-80 < k`

Runtime