Derivation

Split input into 3 regimes
if (* F F) < 0.0
1. Initial program 61.3
  \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
2. Initial simplification61.3
  \[\leadsto (\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]
3. Taylor expanded around inf 61.3
  \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
4. Using strategy rm
5. Applied add-cube-cbrt61.3
  \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)}}\]
6. Using strategy rm
7. Applied add-cube-cbrt61.4
  \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\pi \cdot \ell}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right) \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right)}\right)}\]
8. Using strategy rm
9. Applied add-exp-log61.4
  \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{e^{\log \left(\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\pi \cdot \ell}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right) \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right)\right)\right)}}}\]
10. Applied pow-to-exp61.7
  \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{e^{\log F \cdot 2}} \cdot e^{\log \left(\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\pi \cdot \ell}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right) \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right)\right)\right)}}\]
11. Applied prod-exp61.7
  \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{e^{\log F \cdot 2 + \log \left(\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\pi \cdot \ell}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right) \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right)\right)\right)}}}\]
12. Applied add-exp-log61.7
  \[\leadsto \pi \cdot \ell - \frac{\color{blue}{e^{\log \left(\sin \left(\pi \cdot \ell\right)\right)}}}{e^{\log F \cdot 2 + \log \left(\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\pi \cdot \ell}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right) \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right)\right)\right)}}\]
13. Applied div-exp56.9
  \[\leadsto \pi \cdot \ell - \color{blue}{e^{\log \left(\sin \left(\pi \cdot \ell\right)\right) - \left(\log F \cdot 2 + \log \left(\cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\pi \cdot \ell}} \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right) \cdot \sqrt[3]{\sqrt[3]{\pi \cdot \ell}}\right)\right)\right)\right)}}\]
14. Simplified56.9
  \[\leadsto \pi \cdot \ell - e^{\color{blue}{(-2 \cdot \left(\log F\right) + \left(\log \left(\sin \left(\pi \cdot \ell\right)\right) - \log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}}\]
if 0.0 < (* F F) < 4.77855229072154e-188
1. Initial program 28.9
  \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
2. Initial simplification28.9
  \[\leadsto (\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]
3. Taylor expanded around inf 27.1
  \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
4. Taylor expanded around 0 22.1
  \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}\]
5. Simplified22.1
  \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}}\]
if 4.77855229072154e-188 < (* F F)
1. Initial program 2.3
  \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
2. Initial simplification2.3
  \[\leadsto (\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]
3. Taylor expanded around inf 2.3
  \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
4. Using strategy rm
5. Applied add-log-exp2.3
  \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{\log \left(e^{\cos \left(\pi \cdot \ell\right)}\right)}}\]
Recombined 3 regimes into one program.
Final simplification14.7
\[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \le 0.0:\\ \;\;\;\;\pi \cdot \ell - e^{(-2 \cdot \left(\log F\right) + \left(\log \left(\sin \left(\pi \cdot \ell\right)\right) - \log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}\\ \mathbf{elif}\;F \cdot F \le 4.77855229072154 \cdot 10^{-188}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{(\left({\ell}^{4}\right) \cdot \left(\frac{1}{24} \cdot {\pi}^{4}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_* \cdot {F}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \log \left(e^{\cos \left(\pi \cdot \ell\right)}\right)}\\ \end{array}\]

Error

Derivation

`if (* F F) < 0.0`

`if 0.0 < (* F F) < 4.77855229072154e-188`

`if 4.77855229072154e-188 < (* F F)`

Runtime