Derivation

Split input into 2 regimes
if t < -2.3953048845751785e-47 or 1.7093684075383152e-148 < t
1. Initial program 42.7
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied add-cbrt-cube44.6
  \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
4. Applied simplify30.7
  \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}^{3}}}}\]
5. Using strategy rm
6. Applied associate-*r/30.7
  \[\leadsto \frac{2}{\sqrt[3]{{\left(\left(\tan k \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \color{blue}{\frac{\frac{\sin k}{\frac{\ell}{t}} \cdot t}{\frac{\ell}{t}}}\right)}^{3}}}\]
7. Applied frac-times42.8
  \[\leadsto \frac{2}{\sqrt[3]{{\left(\left(\tan k \cdot \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \frac{\frac{\sin k}{\frac{\ell}{t}} \cdot t}{\frac{\ell}{t}}\right)}^{3}}}\]
8. Applied associate-*r/43.0
  \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{\tan k \cdot \left(k \cdot k\right)}{t \cdot t}} \cdot \frac{\frac{\sin k}{\frac{\ell}{t}} \cdot t}{\frac{\ell}{t}}\right)}^{3}}}\]
9. Applied frac-times41.9
  \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left(\tan k \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right)}{\left(t \cdot t\right) \cdot \frac{\ell}{t}}\right)}}^{3}}}\]
10. Applied cube-div45.6
  \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left(\tan k \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right)\right)}^{3}}{{\left(\left(t \cdot t\right) \cdot \frac{\ell}{t}\right)}^{3}}}}}\]
11. Applied cbrt-div45.0
  \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left(\tan k \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right)\right)}^{3}}}{\sqrt[3]{{\left(\left(t \cdot t\right) \cdot \frac{\ell}{t}\right)}^{3}}}}}\]
12. Applied simplify41.2
  \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\left(k \cdot t\right) \cdot \tan k\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}}{\sqrt[3]{{\left(\left(t \cdot t\right) \cdot \frac{\ell}{t}\right)}^{3}}}}\]
13. Applied simplify17.9
  \[\leadsto \frac{2}{\frac{\left(\left(\left(k \cdot t\right) \cdot \tan k\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\color{blue}{t \cdot \ell}}}\]
14. Using strategy rm
15. Applied clear-num17.9
  \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{t \cdot \ell}{\left(\left(\left(k \cdot t\right) \cdot \tan k\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}}}}\]
16. Applied simplify4.7
  \[\leadsto \frac{2}{\frac{1}{\color{blue}{\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{\tan k}}{k \cdot t}}}}\]
17. Using strategy rm
18. Applied *-un-lft-identity4.7
  \[\leadsto \frac{2}{\frac{1}{\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\color{blue}{1 \cdot \frac{\ell}{\tan k}}}{k \cdot t}}}\]
19. Applied times-frac1.8
  \[\leadsto \frac{2}{\frac{1}{\frac{\frac{\ell}{k}}{\sin k} \cdot \color{blue}{\left(\frac{1}{k} \cdot \frac{\frac{\ell}{\tan k}}{t}\right)}}}\]
if -2.3953048845751785e-47 < t < 1.7093684075383152e-148
1. Initial program 57.7
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied add-cbrt-cube58.2
  \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right) \cdot \left(\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}}\]
4. Applied simplify44.8
  \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left(\tan k \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}\right)\right)}^{3}}}}\]
5. Using strategy rm
6. Applied associate-*r/44.8
  \[\leadsto \frac{2}{\sqrt[3]{{\left(\left(\tan k \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)\right) \cdot \color{blue}{\frac{\frac{\sin k}{\frac{\ell}{t}} \cdot t}{\frac{\ell}{t}}}\right)}^{3}}}\]
7. Applied frac-times45.7
  \[\leadsto \frac{2}{\sqrt[3]{{\left(\left(\tan k \cdot \color{blue}{\frac{k \cdot k}{t \cdot t}}\right) \cdot \frac{\frac{\sin k}{\frac{\ell}{t}} \cdot t}{\frac{\ell}{t}}\right)}^{3}}}\]
8. Applied associate-*r/45.9
  \[\leadsto \frac{2}{\sqrt[3]{{\left(\color{blue}{\frac{\tan k \cdot \left(k \cdot k\right)}{t \cdot t}} \cdot \frac{\frac{\sin k}{\frac{\ell}{t}} \cdot t}{\frac{\ell}{t}}\right)}^{3}}}\]
9. Applied frac-times43.1
  \[\leadsto \frac{2}{\sqrt[3]{{\color{blue}{\left(\frac{\left(\tan k \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right)}{\left(t \cdot t\right) \cdot \frac{\ell}{t}}\right)}}^{3}}}\]
10. Applied cube-div45.0
  \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{\frac{{\left(\left(\tan k \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right)\right)}^{3}}{{\left(\left(t \cdot t\right) \cdot \frac{\ell}{t}\right)}^{3}}}}}\]
11. Applied cbrt-div44.8
  \[\leadsto \frac{2}{\color{blue}{\frac{\sqrt[3]{{\left(\left(\tan k \cdot \left(k \cdot k\right)\right) \cdot \left(\frac{\sin k}{\frac{\ell}{t}} \cdot t\right)\right)}^{3}}}{\sqrt[3]{{\left(\left(t \cdot t\right) \cdot \frac{\ell}{t}\right)}^{3}}}}}\]
12. Applied simplify37.8
  \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\left(k \cdot t\right) \cdot \tan k\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}}{\sqrt[3]{{\left(\left(t \cdot t\right) \cdot \frac{\ell}{t}\right)}^{3}}}}\]
13. Applied simplify19.9
  \[\leadsto \frac{2}{\frac{\left(\left(\left(k \cdot t\right) \cdot \tan k\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}{\color{blue}{t \cdot \ell}}}\]
14. Using strategy rm
15. Applied clear-num19.9
  \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{t \cdot \ell}{\left(\left(\left(k \cdot t\right) \cdot \tan k\right) \cdot \left(k \cdot t\right)\right) \cdot \frac{\sin k}{\ell}}}}}\]
16. Applied simplify2.5
  \[\leadsto \frac{2}{\frac{1}{\color{blue}{\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{\tan k}}{k \cdot t}}}}\]
17. Using strategy rm
18. Applied associate-*l/2.6
  \[\leadsto \frac{2}{\frac{1}{\color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{\tan k}}{k \cdot t}}{\sin k}}}}\]
19. Applied associate-/r/2.6
  \[\leadsto \frac{2}{\color{blue}{\frac{1}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{\tan k}}{k \cdot t}} \cdot \sin k}}\]
20. Applied associate-/r*2.6
  \[\leadsto \color{blue}{\frac{\frac{2}{\frac{1}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{\tan k}}{k \cdot t}}}}{\sin k}}\]
21. Applied simplify2.3
  \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\frac{k}{\ell}}}{\tan k} \cdot \frac{\ell}{k \cdot t}}}{\sin k}\]
Recombined 2 regimes into one program.

Error

Derivation

`if t < -2.3953048845751785e-47 or 1.7093684075383152e-148 < t`

`if -2.3953048845751785e-47 < t < 1.7093684075383152e-148`

Runtime