Average Error: 16.1 → 12.7
Time: 58.1s
Precision: 64
Internal Precision: 2624
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -4.276840822711777 \cdot 10^{+153}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;\ell \le -2.283904777567896 \cdot 10^{-147}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot (\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{F} \cdot \frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{F}\right) \cdot \frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{\cos \left(\pi \cdot \ell\right)}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if l < -4.276840822711777e+153

    1. Initial program 20.3

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

    2. Initial simplification20.3

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

    4. Applied associate-/r*20.3

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

    if -4.276840822711777e+153 < l < -2.283904777567896e-147

    1. Initial program 16.0

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

    2. Initial simplification16.0

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

    3. Taylor expanded around -inf 15.9

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

    4. Taylor expanded around 0 10.7

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{\left(\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)}}\]

    5. Simplified10.7

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \color{blue}{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}}\]

    if -2.283904777567896e-147 < l

    1. Initial program 15.2

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

    2. Initial simplification15.2

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

    3. Taylor expanded around -inf 14.8

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

    5. Applied add-log-exp14.8

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

    7. Applied log1p-expm1-u14.8

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\log_* (1 + (e^{\sin \left(\pi \cdot \ell\right)} - 1)^*)}}{{F}^{2} \cdot \log \left(e^{\cos \left(\pi \cdot \ell\right)}\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt15.0

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

    10. Applied times-frac15.0

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

    11. Simplified11.9

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

    12. Simplified11.9

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

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -4.276840822711777 \cdot 10^{+153}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{\frac{-1}{F}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;\ell \le -2.283904777567896 \cdot 10^{-147}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot (\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \left(\frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{F} \cdot \frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{F}\right) \cdot \frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{\cos \left(\pi \cdot \ell\right)}\\ \end{array}\]

Runtime

Time bar (total: 58.1s)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))