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Derivation

1. Initial program 47.5

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

2. Initial simplification30.7

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

4. Applied times-frac30.6

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

5. Applied times-frac19.9

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

6. Simplified19.3

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

8. Applied *-un-lft-identity19.3

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

9. Applied *-un-lft-identity19.3

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

10. Applied times-frac18.4

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

11. Applied times-frac12.4

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

12. Applied associate-*r*11.1

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

13. Simplified6.8

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

15. Applied associate-*l/6.2

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

16. Simplified1.0

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

18. Applied add-cube-cbrt1.3

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

19. Applied times-frac1.0

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

20. Final simplification1.0

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

Reproduce

herbie shell --seed 2018365 +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))))

Details

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