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Derivation

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 simplification25.4

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

4. Applied add-sqr-sqrt25.5

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

5. Applied times-frac18.9

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

6. Applied times-frac17.0

   \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
7. Using strategy rm

8. Applied add-sqr-sqrt17.0

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

9. Applied sqrt-prod17.0

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

10. Applied *-un-lft-identity17.0

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]

11. Applied times-frac14.2

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]

12. Applied times-frac13.2

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

13. Simplified13.2

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

15. Applied add-sqr-sqrt13.2

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

16. Applied sqrt-prod13.2

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

17. Applied div-inv13.2

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

18. Applied times-frac12.9

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

20. Applied add-cube-cbrt12.9

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

21. Applied sqrt-prod12.9

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

22. Simplified12.9

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

23. Final simplification12.9

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018295 
(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))))