Error

Derivation

1. Split input into 3 regimes

2. if l < -2.4241402976342506e+120

   1. Initial program 21.1

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

   2. Initial simplification21.1

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

   4. Applied associate-/r*21.1

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

   5. Taylor expanded around -inf 21.1

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

   7. Applied *-un-lft-identity21.1

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

   8. Applied associate-/l*21.1

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

   10. Applied add-cube-cbrt21.1

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

   if -2.4241402976342506e+120 < l < 3.021069784627183e+159

   1. Initial program 14.1

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

   2. Initial simplification14.1

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

   4. Applied associate-/r*14.1

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

   5. Taylor expanded around -inf 13.7

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

   6. Taylor expanded around 0 11.4

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}\]

   7. Simplified11.4

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}}\]

   if 3.021069784627183e+159 < l

   1. Initial program 19.5

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

   2. Initial simplification19.5

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

   4. Applied associate-/r*19.5

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

   6. Applied add-cube-cbrt19.4

      \[\leadsto (\left(\tan \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)}\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\]
3. Recombined 3 regimes into one program.

4. Final simplification14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.4241402976342506 \cdot 10^{+120}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(\sqrt[3]{\frac{\cos \left(\pi \cdot \ell\right) \cdot {F}^{2}}{\sin \left(\pi \cdot \ell\right)}} \cdot \sqrt[3]{\frac{\cos \left(\pi \cdot \ell\right) \cdot {F}^{2}}{\sin \left(\pi \cdot \ell\right)}}\right) \cdot \sqrt[3]{\frac{\cos \left(\pi \cdot \ell\right) \cdot {F}^{2}}{\sin \left(\pi \cdot \ell\right)}}}\\ \mathbf{elif}\;\ell \le 3.021069784627183 \cdot 10^{+159}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{(\left({\ell}^{4}\right) \cdot \left(\frac{1}{24} \cdot {\pi}^{4}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_* \cdot {F}^{2}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right)\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \end{array}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018286 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))