Error

Derivation

1. Split input into 3 regimes

2. if l < -1.172518816367358e+152

   1. Initial program 20.0

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

   3. Applied pow220.0

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

   4. Applied pow-flip20.0

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

   5. Simplified20.0

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

   if -1.172518816367358e+152 < l < 1.011634592463237e+172

   1. Initial program 14.7

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

   3. Applied tan-quot14.7

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

   4. Applied frac-times14.4

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

   5. Simplified14.4

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

   7. Applied add-log-exp14.4

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

   8. Taylor expanded around 0 11.8

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

   9. Simplified11.8

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

   if 1.011634592463237e+172 < l

   1. Initial program 20.1

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

   3. Applied add-cube-cbrt20.0

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

4. Final simplification13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.172518816367358 \cdot 10^{+152}:\\ \;\;\;\;\pi \cdot \ell - {F}^{-2} \cdot \tan \left(\pi \cdot \ell\right)\\ \mathbf{elif}\;\ell \le 1.011634592463237 \cdot 10^{+172}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\left(F \cdot F\right) \cdot \log \left(e^{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \tan \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot \frac{1}{F \cdot F}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018263 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))