Error

Derivation

1. Initial program 31.1

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

2. Initial simplification24.4

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

4. Applied *-un-lft-identity24.4

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

5. Applied tan-quot24.4

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

6. Applied associate-*r/24.4

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

7. Applied associate-/r/24.4

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

8. Applied times-frac24.4

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

9. Simplified15.9

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

11. Applied *-un-lft-identity15.9

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

12. Applied times-frac14.2

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

13. Applied associate-*l*12.6

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

14. Simplified12.5

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

16. Applied *-un-lft-identity12.5

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

17. Applied add-cube-cbrt12.8

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

18. Applied times-frac12.8

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

19. Applied add-cube-cbrt12.8

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

20. Applied times-frac12.8

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

21. Applied associate-*l*11.5

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

22. Simplified11.5

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

23. Final simplification11.5

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

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed 2018354 +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))))