Average Error: 15.8 → 14.2
Time: 4.2m
Precision: 64
Internal Precision: 5184
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;F \cdot F \le 0.0:\\ \;\;\;\;\pi \cdot \ell - e^{\log \left(\tan \left(\pi \cdot \ell\right)\right) - \left(\log F + \log F\right)}\\ \mathbf{elif}\;F \cdot F \le 4.2511927412425654 \cdot 10^{-198}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_* \cdot \left(F \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{-1}}{\sqrt[3]{\frac{-1}{\ell}}} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot \left(F \cdot F\right)}\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* F F) < 0.0

    1. Initial program 61.4

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied add-exp-log61.4

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

    4. Applied add-exp-log61.4

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

    5. Applied prod-exp61.4

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

    6. Simplified56.7

      \[\leadsto \pi \cdot \ell - e^{\color{blue}{\log \left(\tan \left(\pi \cdot \ell\right)\right) - \left(\log F + \log F\right)}}\]

    if 0.0 < (* F F) < 4.2511927412425654e-198

    1. Initial program 31.6

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied tan-quot31.6

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

    4. Applied frac-times28.9

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

    5. Simplified28.9

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

    6. Taylor expanded around 0 28.9

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

    7. Taylor expanded around 0 24.9

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

    8. Simplified24.9

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

    if 4.2511927412425654e-198 < (* F F)

    1. Initial program 2.5

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
    2. Using strategy rm

    3. Applied tan-quot2.5

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

    4. Applied frac-times2.5

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

    5. Simplified2.5

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

    6. Taylor expanded around 0 2.5

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

    8. Applied add-cube-cbrt2.5

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

    9. Taylor expanded around -inf 31.5

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

    10. Simplified2.5

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

  4. Final simplification14.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \le 0.0:\\ \;\;\;\;\pi \cdot \ell - e^{\log \left(\tan \left(\pi \cdot \ell\right)\right) - \left(\log F + \log F\right)}\\ \mathbf{elif}\;F \cdot F \le 4.2511927412425654 \cdot 10^{-198}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{(\left({\ell}^{4}\right) \cdot \left({\pi}^{4} \cdot \frac{1}{24}\right) + \left((\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{-1}{2} + 1)_*\right))_* \cdot \left(F \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{-1}}{\sqrt[3]{\frac{-1}{\ell}}} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot \left(F \cdot F\right)}\\ \end{array}\]

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed 2018257 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))