Error

Try it out

Derivation

1. Initial program 47.4

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

2. Initial simplification31.0

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

4. Applied times-frac30.0

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

5. Applied add-cube-cbrt30.1

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

6. Applied times-frac29.7

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

7. Applied times-frac18.8

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

8. Simplified11.3

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

10. Applied div-inv11.3

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

11. Applied *-un-lft-identity11.3

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

12. Applied times-frac11.3

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

13. Applied times-frac7.4

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

14. Simplified7.4

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

15. Simplified7.3

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

17. Applied pow17.3

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

18. Applied pow17.3

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

19. Applied pow-prod-down7.3

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

20. Simplified2.3

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

22. Applied *-un-lft-identity2.3

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

23. Applied associate-/l*2.4

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

24. Final simplification2.4

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

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018258 
(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))))