Average Error: 16.0 → 9.9
Time: 2.5m
Precision: 64
Internal Precision: 4160
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -3.078216244924181 \cdot 10^{-29}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot {F}^{2}}\\ \mathbf{elif}\;F \le -2.1786121626124327 \cdot 10^{-160} \lor \neg \left(F \le 3.999174612283781 \cdot 10^{-163}\right) \land F \le 1.6826208307696995 \cdot 10^{-35}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{(\left(\pi \cdot \ell\right) \cdot \left(\left(F \cdot F\right) \cdot \frac{-1}{3}\right) + \left(\frac{F \cdot F}{\pi \cdot \ell}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot F\right)}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if F < -3.078216244924181e-29

    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 add-cube-cbrt0.4

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

    if -3.078216244924181e-29 < F < -2.1786121626124327e-160 or 3.999174612283781e-163 < F < 1.6826208307696995e-35

    1. Initial program 19.9

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

    2. Initial simplification19.9

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

    3. Taylor expanded around inf 18.9

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

    5. Applied *-un-lft-identity18.9

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

    6. Applied associate-/l*18.9

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

    7. Taylor expanded around 0 9.5

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

    8. Simplified9.5

      \[\leadsto \pi \cdot \ell - \frac{1}{\color{blue}{(\left(\pi \cdot \ell\right) \cdot \left(\left(F \cdot F\right) \cdot \frac{-1}{3}\right) + \left(\frac{F \cdot F}{\pi \cdot \ell}\right))_*}}\]

    if -2.1786121626124327e-160 < F < 3.999174612283781e-163 or 1.6826208307696995e-35 < F

    1. Initial program 23.6

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

    2. Initial simplification23.6

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

    3. Taylor expanded around inf 23.6

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

    5. Applied *-un-lft-identity23.6

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

    6. Applied associate-/l*23.6

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

    8. Applied *-un-lft-identity23.6

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

    9. Applied times-frac23.6

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

    10. Simplified23.6

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

    12. Applied associate-*l*15.8

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -3.078216244924181 \cdot 10^{-29}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot {F}^{2}}\\ \mathbf{elif}\;F \le -2.1786121626124327 \cdot 10^{-160} \lor \neg \left(F \le 3.999174612283781 \cdot 10^{-163}\right) \land F \le 1.6826208307696995 \cdot 10^{-35}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{(\left(\pi \cdot \ell\right) \cdot \left(\left(F \cdot F\right) \cdot \frac{-1}{3}\right) + \left(\frac{F \cdot F}{\pi \cdot \ell}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot \left(\frac{\cos \left(\pi \cdot \ell\right)}{\sin \left(\pi \cdot \ell\right)} \cdot F\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018249 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))