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Derivation

1. Initial program 16.3

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

2. Initial simplification16.1

   \[\leadsto \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\]
3. Using strategy rm

4. Applied *-un-lft-identity16.1

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot \tan \left(\pi \cdot \ell\right)}}{F \cdot F}\]

5. Applied times-frac12.4

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{F} \cdot \frac{\tan \left(\pi \cdot \ell\right)}{F}}\]
6. Using strategy rm

7. Applied *-un-lft-identity12.4

   \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{\color{blue}{1 \cdot \tan \left(\pi \cdot \ell\right)}}{F}\]

8. Applied associate-/l*12.4

   \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \color{blue}{\frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}}\]

9. Taylor expanded around 0 8.0

   \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{1}{\color{blue}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \left(\pi \cdot \ell\right)\right)}}\]
10. Using strategy rm

11. Applied add-log-exp0.7

    \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \color{blue}{\log \left(e^{\pi \cdot \ell}\right)}\right)}\]

12. Final simplification0.7

    \[\leadsto \pi \cdot \ell - \frac{1}{F} \cdot \frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right)}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018227 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))