Average Error: 8.1 → 6.3
Time: 2.6m
Precision: 64
Internal Precision: 128
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le 4.1267488285459326 \cdot 10^{-290}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{\left(\left(\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \left(\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right)}\right) \cdot {F}^{2}}{\sin \left(\pi \cdot \ell\right)}}\\ \mathbf{elif}\;F \le 7.4461750090914135 \cdot 10^{-155}:\\ \;\;\;\;\pi \cdot \ell - e^{(-2 \cdot \left(\log F\right) + \left(\log \left(\sin \left(\pi \cdot \ell\right)\right) - \log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if F < 4.1267488285459326e-290

    1. Initial program 8.0

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

    2. Initial simplification8.0

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

    3. Taylor expanded around inf 7.6

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

    5. Applied add-cube-cbrt7.6

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

    7. Applied add-cube-cbrt7.6

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

    9. Applied clear-num7.6

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

    if 4.1267488285459326e-290 < F < 7.4461750090914135e-155

    1. Initial program 60.0

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

    2. Initial simplification60.0

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

    3. Taylor expanded around inf 55.9

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

    5. Applied add-cube-cbrt55.9

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

    7. Applied add-exp-log55.9

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

    8. Applied add-exp-log56.0

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

    9. Applied prod-exp56.0

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

    10. Applied add-exp-log58.0

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

    11. Applied div-exp58.0

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

    12. Simplified34.1

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

    if 7.4461750090914135e-155 < F

    1. Initial program 0.6

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

    2. Initial simplification0.6

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

    4. Applied associate-/r*0.6

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le 4.1267488285459326 \cdot 10^{-290}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{\left(\left(\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \left(\sqrt[3]{\cos \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\cos \left(\pi \cdot \ell\right)}\right)}\right) \cdot {F}^{2}}{\sin \left(\pi \cdot \ell\right)}}\\ \mathbf{elif}\;F \le 7.4461750090914135 \cdot 10^{-155}:\\ \;\;\;\;\pi \cdot \ell - e^{(-2 \cdot \left(\log F\right) + \left(\log \left(\sin \left(\pi \cdot \ell\right)\right) - \log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \end{array}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))