Error

Derivation

1. Initial program 47.4

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

2. Simplified30.4

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

3. Taylor expanded around inf 23.6

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

4. Simplified19.6

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

6. Applied associate-*l/20.6

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

7. Applied associate-/r/20.6

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

8. Applied *-un-lft-identity20.6

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

9. Applied times-frac20.6

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

10. Simplified7.1

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

12. Applied associate-/l*1.6

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

14. Applied *-un-lft-identity1.6

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

15. Applied times-frac1.0

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

16. Simplified1.0

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

17. Final simplification1.0

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

Reproduce

herbie shell --seed 2019068 
(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))))