Error

Try it out

Derivation

1. Initial program 46.9

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

3. Applied add-cube-cbrt46.9

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

4. Applied associate-*r*46.9

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

5. Applied simplify38.1

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

7. Applied add-sqr-sqrt38.1

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

8. Applied cbrt-prod38.1

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

9. Applied simplify38.1

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

10. Applied simplify26.9

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

11. Taylor expanded around 0 27.4

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

12. Applied simplify26.8

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

13. Taylor expanded around 0 27.3

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

14. Applied simplify26.1

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018178 
(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))))