Error

Derivation

1. Split input into 2 regimes

2. if k < -3.395347975146318e-149 or 5.649848822744807e-237 < k

   1. Initial program 31.3

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt31.4

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}\right) \cdot \sqrt[3]{{t}^{3}}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   4. Applied times-frac28.4

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}}{\ell} \cdot \frac{\sqrt[3]{{t}^{3}}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   5. Applied simplify28.4

      \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{{t}^{3}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   6. Applied simplify18.5

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   7. Using strategy rm

   8. Applied tan-quot18.5

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   9. Applied associate-*l/18.5

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot \frac{t}{\ell}}{\frac{\ell}{t}}} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   10. Applied associate-*l/17.4

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k}{\frac{\ell}{t}}} \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   11. Applied frac-times17.3

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}{\frac{\ell}{t} \cdot \cos k}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   12. Applied associate-*l/15.6

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{t} \cdot \cos k}}}\]

   if -3.395347975146318e-149 < k < 5.649848822744807e-237

   1. Initial program 39.7

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt39.7

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}\right) \cdot \sqrt[3]{{t}^{3}}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   4. Applied times-frac37.1

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}}{\ell} \cdot \frac{\sqrt[3]{{t}^{3}}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   5. Applied simplify37.0

      \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{{t}^{3}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   6. Applied simplify31.5

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   7. Using strategy rm

   8. Applied tan-quot31.5

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   9. Applied associate-*l/31.5

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{t \cdot \frac{t}{\ell}}{\frac{\ell}{t}}} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   10. Applied associate-*l/19.9

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k}{\frac{\ell}{t}}} \cdot \frac{\sin k}{\cos k}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   11. Applied frac-times27.0

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}{\frac{\ell}{t} \cdot \cos k}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   12. Applied associate-*l/27.0

       \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{t} \cdot \cos k}}}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt27.2

       \[\leadsto \frac{2}{\frac{\color{blue}{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}{\frac{\ell}{t} \cdot \cos k}}\]
   15. Using strategy rm

   16. Applied flip3-+27.2

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \color{blue}{\frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}}{\frac{\ell}{t} \cdot \cos k}}\]

   17. Applied associate-*r/31.9

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \sqrt[3]{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \sin k\right) \cdot \sin k}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   18. Applied associate-*l/32.4

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \sqrt[3]{\color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}} \cdot \sin k}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   19. Applied associate-*l/37.7

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \sqrt[3]{\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}{\ell}}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   20. Applied cbrt-div37.7

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \color{blue}{\frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   21. Applied associate-*r/37.7

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \sin k\right) \cdot \sin k}\right) \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   22. Applied associate-*l/37.7

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}} \cdot \sin k}\right) \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   23. Applied associate-*l/37.7

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}{\ell}}}\right) \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   24. Applied cbrt-div37.7

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k} \cdot \color{blue}{\frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}}\right) \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   25. Applied associate-*r/37.7

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\left(\color{blue}{\frac{t \cdot t}{\ell}} \cdot \sin k\right) \cdot \sin k} \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   26. Applied associate-*l/35.5

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\color{blue}{\frac{\left(t \cdot t\right) \cdot \sin k}{\ell}} \cdot \sin k} \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   27. Applied associate-*l/35.5

       \[\leadsto \frac{2}{\frac{\left(\left(\sqrt[3]{\color{blue}{\frac{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}{\ell}}} \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   28. Applied cbrt-div35.6

       \[\leadsto \frac{2}{\frac{\left(\left(\color{blue}{\frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   29. Applied frac-times35.6

       \[\leadsto \frac{2}{\frac{\left(\color{blue}{\frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   30. Applied frac-times35.6

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\left(\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot \frac{{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   31. Applied frac-times35.6

       \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\left(\left(\sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k} \cdot \sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \sqrt[3]{\left(\left(t \cdot t\right) \cdot \sin k\right) \cdot \sin k}\right) \cdot \left({\left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}^{3} + {1}^{3}\right)}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)\right)}}}{\frac{\ell}{t} \cdot \cos k}}\]

   32. Applied simplify18.4

       \[\leadsto \frac{2}{\frac{\frac{\color{blue}{(\left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \left({\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right)}^{3}\right) + \left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right))_*}}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(1 \cdot 1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot 1\right)\right)}}{\frac{\ell}{t} \cdot \cos k}}\]

   33. Applied simplify18.1

       \[\leadsto \frac{2}{\frac{\frac{(\left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \left({\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right)}^{3}\right) + \left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right))_*}{\color{blue}{\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) + 1)_* - (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \ell}}}{\frac{\ell}{t} \cdot \cos k}}\]
3. Recombined 2 regimes into one program.

4. Applied simplify15.9

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;k \le -3.395347975146318 \cdot 10^{-149} \lor \neg \left(k \le 5.649848822744807 \cdot 10^{-237}\right):\\ \;\;\;\;\frac{2}{\frac{\left(\sin k \cdot \left(\sin k \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\frac{\ell}{t} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{(\left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \left({\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right)}^{3}\right) + \left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right))_*}{\left((\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) + 1)_* - (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 1)_*\right) \cdot \ell}}{\frac{\ell}{t} \cdot \cos k}}\\ \end{array}}\]

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))