Derivation

Split input into 3 regimes
if (* F F) < 6.1708799165572e-321
1. Initial program 61.4
  \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
2. Initial simplification61.4
  \[\leadsto (\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]
3. Taylor expanded around -inf 61.2
  \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
4. Using strategy rm
5. Applied *-un-lft-identity61.2
  \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot \sin \left(\pi \cdot \ell\right)}}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}\]
6. Applied associate-/l*61.2
  \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{\frac{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]
7. Using strategy rm
8. Applied *-un-lft-identity61.2
  \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}{\color{blue}{1 \cdot \sin \left(\pi \cdot \ell\right)}}}\]
9. Applied times-frac61.2
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{{F}^{2}}{1} \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]
10. Simplified61.2
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(F \cdot F\right)} \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}\]
11. Using strategy rm
12. Applied associate-*l*42.3
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{F \cdot \left(F \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}\right)}}\]
if 6.1708799165572e-321 < (* F F) < 6.916689803841113e-57
1. Initial program 20.6
  \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
2. Initial simplification20.6
  \[\leadsto (\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]
3. Taylor expanded around -inf 19.7
  \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
4. Using strategy rm
5. Applied *-un-lft-identity19.7
  \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot \sin \left(\pi \cdot \ell\right)}}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}\]
6. Applied associate-/l*19.7
  \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{\frac{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]
7. Using strategy rm
8. Applied *-un-lft-identity19.7
  \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}{\color{blue}{1 \cdot \sin \left(\pi \cdot \ell\right)}}}\]
9. Applied times-frac19.7
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{{F}^{2}}{1} \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]
10. Simplified19.7
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(F \cdot F\right)} \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}\]
11. Taylor expanded around 0 9.6
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{{F}^{2}}{\pi \cdot \ell} - \frac{1}{3} \cdot \left({F}^{2} \cdot \left(\pi \cdot \ell\right)\right)}}\]
12. Simplified9.6
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{(\left(\pi \cdot \ell\right) \cdot \left(\left(F \cdot F\right) \cdot \frac{-1}{3}\right) + \left(\frac{F \cdot F}{\pi \cdot \ell}\right))_*}}\]
if 6.916689803841113e-57 < (* F F)
1. Initial program 0.4
  \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
2. Initial simplification0.4
  \[\leadsto (\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]
3. Taylor expanded around -inf 0.4
  \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}}\]
4. Using strategy rm
5. Applied *-un-lft-identity0.4
  \[\leadsto \pi \cdot \ell - \frac{\color{blue}{1 \cdot \sin \left(\pi \cdot \ell\right)}}{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}\]
6. Applied associate-/l*0.4
  \[\leadsto \pi \cdot \ell - \color{blue}{\frac{1}{\frac{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]
7. Using strategy rm
8. Applied *-un-lft-identity0.4
  \[\leadsto \pi \cdot \ell - \frac{1}{\frac{{F}^{2} \cdot \cos \left(\pi \cdot \ell\right)}{\color{blue}{1 \cdot \sin \left(\pi \cdot \ell\right)}}}\]
9. Applied times-frac0.5
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\frac{{F}^{2}}{1} \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]
10. Simplified0.5
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\left(F \cdot F\right)} \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}\]
11. Using strategy rm
12. Applied add-cbrt-cube1.9
  \[\leadsto \pi \cdot \ell - \frac{1}{\left(F \cdot F\right) \cdot \color{blue}{\sqrt[3]{\left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}\right) \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}}\]
13. Applied add-cbrt-cube1.9
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\sqrt[3]{\left(\left(F \cdot F\right) \cdot \left(F \cdot F\right)\right) \cdot \left(F \cdot F\right)}} \cdot \sqrt[3]{\left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}\right) \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}}}\]
14. Applied cbrt-unprod1.9
  \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{\sqrt[3]{\left(\left(\left(F \cdot F\right) \cdot \left(F \cdot F\right)\right) \cdot \left(F \cdot F\right)\right) \cdot \left(\left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}\right) \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}\right)}}}\]
15. Simplified1.8
  \[\leadsto \pi \cdot \ell - \frac{1}{\sqrt[3]{\color{blue}{\frac{\frac{\left(F \cdot {F}^{5}\right) \cdot \cos \left(\pi \cdot \ell\right)}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)} \cdot \sin \left(\pi \cdot \ell\right)}}}}\]
Recombined 3 regimes into one program.
Final simplification10.9
\[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \le 6.1708799165572 \cdot 10^{-321}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\left(F \cdot \frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)}\right) \cdot F}\\ \mathbf{elif}\;F \cdot F \le 6.916689803841113 \cdot 10^{-57}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{(\left(\pi \cdot \ell\right) \cdot \left(\frac{-1}{3} \cdot \left(F \cdot F\right)\right) + \left(\frac{F \cdot F}{\pi \cdot \ell}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\sqrt[3]{\frac{\frac{\cos \left(\pi \cdot \ell\right) \cdot \left(F \cdot {F}^{5}\right)}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)} \cdot \sin \left(\pi \cdot \ell\right)}}}\\ \end{array}\]

Error

Derivation

`if (* F F) < 6.1708799165572e-321`

`if 6.1708799165572e-321 < (* F F) < 6.916689803841113e-57`

`if 6.916689803841113e-57 < (* F F)`

Runtime