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Derivation

1. Initial program 32.1

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

2. Initial simplification24.2

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

4. Applied add-sqr-sqrt24.2

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

5. Applied times-frac18.2

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

6. Applied times-frac16.5

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

8. Applied add-sqr-sqrt16.5

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

9. Applied sqrt-prod16.5

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

10. Applied *-un-lft-identity16.5

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

11. Applied times-frac13.4

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

12. Applied times-frac12.3

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

13. Simplified12.3

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

14. Final simplification12.3

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018221 
(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))))