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Derivation

1. Initial program 8.4

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

2. Simplified8.4

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

3. Taylor expanded around -inf 8.0

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
4. Using strategy rm

5. Applied expm1-log1p-u8.0

   \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{(e^{\log_* (1 + \cos \left(\pi \cdot \ell\right))} - 1)^*}}\]
6. Using strategy rm

7. Applied clear-num8.0

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{\frac{{F}^{2} \cdot (e^{\log_* (1 + \cos \left(\pi \cdot \ell\right))} - 1)^*}{\sin \left(\pi \cdot \ell\right)}}}\]
8. Using strategy rm

9. Applied *-un-lft-identity8.0

   \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{1 \cdot \frac{{F}^{2} \cdot (e^{\log_* (1 + \cos \left(\pi \cdot \ell\right))} - 1)^*}{\sin \left(\pi \cdot \ell\right)}}}\]

10. Applied associate-/r*8.0

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{1}}{\frac{{F}^{2} \cdot (e^{\log_* (1 + \cos \left(\pi \cdot \ell\right))} - 1)^*}{\sin \left(\pi \cdot \ell\right)}}}\]

11. Simplified0.6

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

12. Final simplification0.6

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

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))

Details

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