Average Error: 16.0 → 14.1
Time: 1.3m
Precision: 64
Internal Precision: 2880
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.728330292434146 \cdot 10^{+152}:\\ \;\;\;\;(\left(\log_* (1 + (e^{\tan \left(\pi \cdot \ell\right)} - 1)^*)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;\pi \cdot \ell \le 4.0451749976185865 \cdot 10^{+156}:\\ \;\;\;\;(\left(\frac{\sin \left(\pi \cdot \ell\right)}{(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)}\right) \cdot \left(\left(\sqrt[3]{\frac{-1}{F \cdot F}} \cdot \sqrt[3]{\frac{-1}{F \cdot F}}\right) \cdot \sqrt[3]{\frac{-1}{F \cdot F}}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{\sqrt[3]{-1}}{F} \cdot \frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -2.728330292434146e+152

    1. Initial program 20.8

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

    2. Initial simplification20.8

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

    4. Applied log1p-expm1-u21.2

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

    if -2.728330292434146e+152 < (* PI l) < 4.0451749976185865e+156

    1. Initial program 14.4

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

    2. Initial simplification14.4

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

    4. Applied add-cube-cbrt14.6

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

    6. Applied tan-quot14.6

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

    7. Taylor expanded around 0 11.7

      \[\leadsto (\left(\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{\left(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - \frac{1}{2} \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)}}\right) \cdot \left(\left(\sqrt[3]{\frac{-1}{F \cdot F}} \cdot \sqrt[3]{\frac{-1}{F \cdot F}}\right) \cdot \sqrt[3]{\frac{-1}{F \cdot F}}\right) + \left(\pi \cdot \ell\right))_*\]

    8. Simplified11.7

      \[\leadsto (\left(\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{2}}}\right) \cdot \left(\left(\sqrt[3]{\frac{-1}{F \cdot F}} \cdot \sqrt[3]{\frac{-1}{F \cdot F}}\right) \cdot \sqrt[3]{\frac{-1}{F \cdot F}}\right) + \left(\pi \cdot \ell\right))_*\]

    if 4.0451749976185865e+156 < (* PI l)

    1. Initial program 19.6

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

    2. Initial simplification19.6

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

    4. Applied add-cube-cbrt19.6

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

    5. Applied times-frac19.6

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

  4. Final simplification14.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -2.728330292434146 \cdot 10^{+152}:\\ \;\;\;\;(\left(\log_* (1 + (e^{\tan \left(\pi \cdot \ell\right)} - 1)^*)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{elif}\;\pi \cdot \ell \le 4.0451749976185865 \cdot 10^{+156}:\\ \;\;\;\;(\left(\frac{\sin \left(\pi \cdot \ell\right)}{(\frac{1}{24} \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1)_* - \frac{1}{2} \cdot \left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)}\right) \cdot \left(\left(\sqrt[3]{\frac{-1}{F \cdot F}} \cdot \sqrt[3]{\frac{-1}{F \cdot F}}\right) \cdot \sqrt[3]{\frac{-1}{F \cdot F}}\right) + \left(\pi \cdot \ell\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\tan \left(\pi \cdot \ell\right)\right) \cdot \left(\frac{\sqrt[3]{-1}}{F} \cdot \frac{\sqrt[3]{-1} \cdot \sqrt[3]{-1}}{F}\right) + \left(\pi \cdot \ell\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018217 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))