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Derivation

1. Initial program 47.3

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

2. Initial simplification30.7

   \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
3. Using strategy rm

4. Applied times-frac29.8

   \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

5. Applied add-cube-cbrt29.8

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

6. Applied times-frac29.4

   \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

7. Applied times-frac18.2

   \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]

8. Simplified10.9

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

10. Applied div-inv11.0

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

11. Applied add-cube-cbrt11.1

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

12. Applied times-frac11.1

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

13. Applied times-frac7.3

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

14. Simplified7.3

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

16. Applied *-un-lft-identity7.3

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

17. Applied add-cube-cbrt7.3

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

18. Applied times-frac7.3

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

19. Applied cbrt-prod7.3

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

20. Simplified7.3

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

22. Applied div-inv7.3

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

23. Applied add-cube-cbrt7.3

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

24. Applied times-frac7.3

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

25. Applied cbrt-prod7.3

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

26. Final simplification7.3

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018274 
(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))))