Error

Derivation

1. Initial program 15.9

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

2. Applied simplify15.7

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

4. Applied associate-/r*11.9

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

6. Applied clear-num11.9

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

7. Taylor expanded around 0 31.2

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

8. Applied simplify8.3

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

10. Applied add-cube-cbrt8.3

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))