Error

Derivation

1. Initial program 32.6

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

2. Initial simplification25.1

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

4. Applied *-un-lft-identity25.1

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

5. Applied tan-quot25.1

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

6. Applied associate-*r/25.1

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

7. Applied associate-/r/25.1

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

8. Applied times-frac25.1

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

9. Simplified16.9

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

11. Applied add-cube-cbrt17.1

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

12. Applied times-frac15.2

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

13. Applied associate-*l*12.4

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

15. Applied *-un-lft-identity12.4

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

16. Applied add-cube-cbrt12.4

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

17. Applied cbrt-prod12.4

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

18. Applied times-frac12.4

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

19. Applied associate-*l*12.5

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

20. Simplified12.5

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

21. Final simplification12.5

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

Runtime

Time bar (total: 5.5m)Debug logProfile

herbie shell --seed 2018248 +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))))