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Derivation

1. Initial program 47.0

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

2. Taylor expanded around -inf 23.5

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

4. Applied associate-/l*23.4

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

6. Applied associate-/r*22.0

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

8. Applied div-inv22.1

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

9. Applied associate-/r*22.0

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

10. Simplified7.2

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

11. Final simplification7.2

    \[\leadsto \frac{\frac{2}{t}}{\frac{\frac{\sin k}{\frac{\ell}{k}} \cdot \frac{\sin k}{\frac{\ell}{k}}}{\cos k}}\]

Reproduce

herbie shell --seed 2019030 +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))))