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Derivation

1. Initial program 46.6

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

2. Initial simplification30.0

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

4. Applied times-frac29.1

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

5. Applied add-cube-cbrt29.2

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

6. Applied times-frac28.9

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

7. Applied times-frac18.0

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

8. Simplified10.7

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

10. Applied *-un-lft-identity10.7

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

11. Applied div-inv10.7

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

12. Applied add-cube-cbrt10.8

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

13. Applied times-frac10.9

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

14. Applied times-frac11.1

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

15. Applied times-frac11.2

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

16. Simplified11.2

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

17. Simplified7.3

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

19. Applied div-inv7.3

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

20. Applied associate-*r*2.7

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

22. Applied tan-quot2.7

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

23. Applied associate-/r/2.7

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

24. Applied cbrt-prod2.7

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

25. Applied times-frac2.7

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

26. Applied associate-*r*2.3

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

27. Final simplification2.3

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018339 
(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))))