Average Error: 16.2 → 13.1
Time: 1.3m
Precision: 64
Internal Precision: 3136
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -0.9247355462199732:\\ \;\;\;\;(\left(\tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;F \le -7.953551036186238 \cdot 10^{-184}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{(\left(\pi \cdot \ell\right) \cdot \left(\frac{-1}{3} \cdot \left(F \cdot F\right)\right) + \left(\frac{F \cdot F}{\pi \cdot \ell}\right))_*}\\ \mathbf{elif}\;F \le 2.7251553284785375 \cdot 10^{-162}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{e^{(2 \cdot \left(\log F\right) + \left(\log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_* - \log \left(\sin \left(\pi \cdot \ell\right)\right)}}\\ \mathbf{elif}\;F \le 3.2620238829726224 \cdot 10^{-13}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{(\left(\pi \cdot \ell\right) \cdot \left(\frac{-1}{3} \cdot \left(F \cdot F\right)\right) + \left(\frac{F \cdot F}{\pi \cdot \ell}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\log_* (1 + (e^{\tan \left(\pi \cdot \ell\right)} - 1)^*)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 4 regimes

  2. if F < -0.9247355462199732

    1. Initial program 0.2

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

    2. Initial simplification0.2

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

    4. Applied associate-/r*0.2

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

    6. Applied add-cube-cbrt0.2

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

    if -0.9247355462199732 < F < -7.953551036186238e-184 or 2.7251553284785375e-162 < F < 3.2620238829726224e-13

    1. Initial program 20.7

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

    2. Initial simplification20.7

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

    3. Taylor expanded around inf 19.8

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

    5. Applied *-un-lft-identity19.8

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

    6. Applied associate-/l*19.8

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

    7. Taylor expanded around 0 12.7

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

    8. Simplified12.7

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

    if -7.953551036186238e-184 < F < 2.7251553284785375e-162

    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 61.4

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

    5. Applied *-un-lft-identity61.4

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

    6. Applied associate-/l*61.4

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

    8. Applied add-exp-log61.4

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

    9. Applied add-exp-log61.4

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

    10. Applied add-exp-log61.4

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

    11. Applied prod-exp61.4

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

    12. Applied div-exp61.4

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

    13. Simplified56.0

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

    if 3.2620238829726224e-13 < F

    1. Initial program 0.3

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

    2. Initial simplification0.3

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

    4. Applied log1p-expm1-u0.5

      \[\leadsto (\color{blue}{\left(\log_* (1 + (e^{\tan \left(\pi \cdot \ell\right)} - 1)^*)\right)} \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -0.9247355462199732:\\ \;\;\;\;(\left(\tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;F \le -7.953551036186238 \cdot 10^{-184}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{(\left(\pi \cdot \ell\right) \cdot \left(\frac{-1}{3} \cdot \left(F \cdot F\right)\right) + \left(\frac{F \cdot F}{\pi \cdot \ell}\right))_*}\\ \mathbf{elif}\;F \le 2.7251553284785375 \cdot 10^{-162}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{e^{(2 \cdot \left(\log F\right) + \left(\log \left(\cos \left(\pi \cdot \ell\right)\right)\right))_* - \log \left(\sin \left(\pi \cdot \ell\right)\right)}}\\ \mathbf{elif}\;F \le 3.2620238829726224 \cdot 10^{-13}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{(\left(\pi \cdot \ell\right) \cdot \left(\frac{-1}{3} \cdot \left(F \cdot F\right)\right) + \left(\frac{F \cdot F}{\pi \cdot \ell}\right))_*}\\ \mathbf{else}:\\ \;\;\;\;(\left(\log_* (1 + (e^{\tan \left(\pi \cdot \ell\right)} - 1)^*)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes16.413.111.15.361.6%
herbie shell --seed 2018339 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))