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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. Using strategy rm

3. Applied add-cube-cbrt47.0

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

4. Applied associate-*r*47.0

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

5. Applied simplify40.0

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

7. Applied *-un-lft-identity40.0

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

8. Applied cbrt-prod40.0

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

9. Applied simplify30.0

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

11. Applied times-frac28.9

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

12. Applied associate-*l*27.2

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

14. Applied div-inv27.2

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

15. Applied associate-*l*25.8

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

16. Applied simplify25.8

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed 2020178 
(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))))