Error

Derivation

1. Initial program 8.6

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

2. Simplified8.1

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

4. Applied *-un-lft-identity8.1

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

5. Applied associate-*r*8.1

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

6. Applied *-un-lft-identity8.1

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

7. Applied times-frac0.7

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

8. Simplified0.7

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

10. Applied clear-num0.7

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

11. Applied un-div-inv0.7

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

12. Final simplification0.7

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))