Error

Derivation

1. Initial program 16.3

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

3. Applied associate-*l/16.0

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

4. Simplified16.0

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

6. Applied *-un-lft-identity16.0

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

7. Applied times-frac12.4

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

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

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

10. Applied associate-/l*12.4

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

11. Taylor expanded around 0 8.1

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

12. Simplified8.1

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

13. Final simplification8.1

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))