Average Error: 15.3 → 12.7
Time: 1.2m
Precision: 64
Internal Precision: 2880
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -4.278017048520236 \cdot 10^{+153} \lor \neg \left(\ell \le 1.9745454511546535 \cdot 10^{+152}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \frac{\sqrt[3]{\pi} \cdot \sqrt[3]{-1}}{\sqrt[3]{\frac{-1}{\ell}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\log_* (1 + (e^{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*} - 1)^*) \cdot {F}^{2}}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 2 regimes

  2. if l < -4.278017048520236e+153 or 1.9745454511546535e+152 < l

    1. Initial program 18.4

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

    2. Initial simplification18.4

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

    3. Taylor expanded around inf 18.4

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

    5. Applied add-cube-cbrt18.4

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

    6. Taylor expanded around -inf 38.3

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

    7. Simplified18.4

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

    if -4.278017048520236e+153 < l < 1.9745454511546535e+152

    1. Initial program 14.2

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

    2. Initial simplification14.2

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

    3. Taylor expanded around inf 13.9

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

    5. Applied log1p-expm1-u13.9

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

    6. Taylor expanded around 0 10.6

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

    7. Simplified10.6

      \[\leadsto \pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \log_* (1 + (e^{\color{blue}{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*}} - 1)^*)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -4.278017048520236 \cdot 10^{+153} \lor \neg \left(\ell \le 1.9745454511546535 \cdot 10^{+152}\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{{F}^{2} \cdot \cos \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \frac{\sqrt[3]{\pi} \cdot \sqrt[3]{-1}}{\sqrt[3]{\frac{-1}{\ell}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\log_* (1 + (e^{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_*} - 1)^*) \cdot {F}^{2}}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018273 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))