Derivation

Split input into 3 regimes
if k < -0.05023041948797596
1. Initial program 31.6
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
2. Initial simplification19.7
  \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
3. Using strategy rm
4. Applied times-frac19.7
  \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
5. Applied associate-/l*17.4
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}}\]
6. Taylor expanded around -inf 22.5
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{\sin k \cdot {k}^{2}}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{{\ell}^{2} \cdot \cos k}}}\]
7. Simplified4.9
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{\sin k}{\cos k} \cdot \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}\]
8. Using strategy rm
9. Applied add-sqr-sqrt4.9
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\sin k}{\cos k} \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}} \cdot \sqrt{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}}\right)}}\]
10. Applied associate-*r*4.9
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\left(\frac{\sin k}{\cos k} \cdot \sqrt{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}}\right) \cdot \sqrt{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}}}}\]
11. Using strategy rm
12. Applied *-un-lft-identity4.9
  \[\leadsto \frac{\color{blue}{1 \cdot \frac{\frac{2}{t}}{\sin k}}}{\left(\frac{\sin k}{\cos k} \cdot \sqrt{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}}\right) \cdot \sqrt{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}}}\]
13. Applied associate-/l*5.1
  \[\leadsto \color{blue}{\frac{1}{\frac{\left(\frac{\sin k}{\cos k} \cdot \sqrt{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}}\right) \cdot \sqrt{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}}}{\frac{\frac{2}{t}}{\sin k}}}}\]
if -0.05023041948797596 < k < 6.40322444096831e+29
1. Initial program 33.3
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
2. Initial simplification32.2
  \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
3. Using strategy rm
4. Applied times-frac16.1
  \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
5. Applied associate-/l*15.4
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}}\]
6. Using strategy rm
7. Applied *-un-lft-identity15.4
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}}\]
8. Applied times-frac8.7
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}}\]
9. Applied add-sqr-sqrt8.9
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\color{blue}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}\]
10. Applied times-frac8.4
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{1}} \cdot \frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{\tan k}}}}\]
11. Applied add-cube-cbrt8.7
  \[\leadsto \frac{\frac{\frac{2}{t}}{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{1}} \cdot \frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{\tan k}}}\]
12. Applied *-un-lft-identity8.7
  \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{2}{t}}}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{1}} \cdot \frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{\tan k}}}\]
13. Applied times-frac8.7
  \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{1}} \cdot \frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{\tan k}}}\]
14. Applied times-frac4.7
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{1}}} \cdot \frac{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{\tan k}}}}\]
15. Simplified4.5
  \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}} \cdot \frac{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}{\frac{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\ell}{t}}{\tan k}}}\]
if 6.40322444096831e+29 < k
1. Initial program 32.9
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
2. Initial simplification20.8
  \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
3. Using strategy rm
4. Applied times-frac20.8
  \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
5. Applied associate-/l*18.3
  \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\frac{\frac{k}{t} \cdot \frac{k}{t} + 2}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}}\]
6. Taylor expanded around -inf 24.6
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{\sin k \cdot {k}^{2}}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{2} \cdot \sin k}{{\ell}^{2} \cdot \cos k}}}\]
7. Simplified5.2
  \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\color{blue}{\frac{\sin k}{\cos k} \cdot \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}\]
8. Using strategy rm
9. Applied div-inv5.2
  \[\leadsto \frac{\color{blue}{\frac{2}{t} \cdot \frac{1}{\sin k}}}{\frac{\sin k}{\cos k} \cdot \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\]
10. Applied associate-/l*5.2
  \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{\frac{\sin k}{\cos k} \cdot \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + \frac{k}{\ell} \cdot \frac{k}{\ell}\right)}{\frac{1}{\sin k}}}}\]
Recombined 3 regimes into one program.
Final simplification4.9
\[\leadsto \begin{array}{l} \mathbf{if}\;k \le -0.05023041948797596:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{k}{\ell} \cdot \frac{k}{\ell} + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2} \cdot \left(\frac{\sin k}{\cos k} \cdot \sqrt{\frac{k}{\ell} \cdot \frac{k}{\ell} + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}\right)}{\frac{\frac{2}{t}}{\sin k}}}\\ \mathbf{elif}\;k \le 6.40322444096831 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}{\frac{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}{\frac{\frac{\ell}{t}}{\tan k}}} \cdot \frac{\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{\left(\frac{k}{\ell} \cdot \frac{k}{\ell} + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2\right) \cdot \frac{\sin k}{\cos k}}{\frac{1}{\sin k}}}\\ \end{array}\]

Error

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Derivation

`if k < -0.05023041948797596`

`if -0.05023041948797596 < k < 6.40322444096831e+29`

`if 6.40322444096831e+29 < k`

Runtime

In
Out