Error

Derivation

1. Split input into 2 regimes

2. if F < 6.233122476385808e-248 or 6.426435977887407e-171 < F

   1. Initial program 6.5

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

   2. Simplified6.5

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

   3. Taylor expanded around inf 5.9

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

   if 6.233122476385808e-248 < F < 6.426435977887407e-171

   1. Initial program 60.0

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

   2. Simplified60.0

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

   3. Taylor expanded around inf 60.0

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

   5. Applied add-exp-log60.0

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

   6. Applied add-exp-log60.0

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

   7. Applied prod-exp60.0

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

   8. Applied add-exp-log60.0

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

   9. Applied div-exp60.0

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

   10. Simplified35.6

       \[\leadsto \pi \cdot \ell - e^{\color{blue}{\log \left(\sin \left(\pi \cdot \ell\right)\right) - (2 \cdot \left(\log F\right) + \left(\log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}}\]
3. Recombined 2 regimes into one program.

4. Final simplification7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le 6.233122476385808 \cdot 10^{-248} \lor \neg \left(F \le 6.426435977887407 \cdot 10^{-171}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right) \cdot {F}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - e^{\log \left(\sin \left(\pi \cdot \ell\right)\right) - (2 \cdot \left(\log F\right) + \left(\log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))

Details

Time bar (total: 38.2s)Debug log