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Derivation

1. Initial program 16.6

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

2. Initial simplification16.3

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

4. Applied *-un-lft-identity16.3

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

5. Applied times-frac12.6

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

7. Applied associate-*r/12.6

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

9. Applied clear-num12.6

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

10. Taylor expanded around 0 30.9

    \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{{F}^{2}}{\pi \cdot \ell} - \frac{1}{3} \cdot \left({F}^{2} \cdot \left(\pi \cdot \ell\right)\right)}}\]

11. Simplified8.4

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

12. Final simplification8.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018230 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))