Average Error: 16.0 → 12.6
Time: 1.7m
Precision: 64
Internal Precision: 2880
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \le -797726977.2823014:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)\\ \mathbf{if}\;\pi \cdot \ell \le 4.004813796983293 \cdot 10^{-127}:\\ \;\;\;\;(\left(\frac{-\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right) + (\left(\frac{2}{15} \cdot {\pi}^{5}\right) \cdot \left({\ell}^{5}\right) + \left(\ell \cdot \pi\right))_*}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)} \cdot \sqrt[3]{\tan \left(\ell \cdot \pi\right)}} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right)\right) + \left(\ell \cdot \pi\right))_*\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\sqrt{\pi \cdot \ell} \cdot \sqrt{\pi \cdot \ell}\right)\\ \end{array}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Split input into 3 regimes

  2. if (* PI l) < -797726977.2823014

    1. Initial program 21.9

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

    3. Applied add-cube-cbrt21.9

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

    if -797726977.2823014 < (* PI l) < 4.004813796983293e-127

    1. Initial program 10.1

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

    3. Applied add-cube-cbrt10.4

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

    4. Applied associate-*r*10.4

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

    5. Applied simplify4.1

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

    7. Applied add-cube-cbrt4.1

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

    8. Applied cbrt-prod4.2

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

    10. Applied add-cube-cbrt4.2

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

    11. Taylor expanded around 0 4.4

      \[\leadsto \pi \cdot \ell - \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \frac{\sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)} \cdot \sqrt[3]{\tan \left(\ell \cdot \pi\right)}} \cdot \sqrt[3]{\sqrt[3]{\color{blue}{\frac{1}{3} \cdot \left({\pi}^{3} \cdot {\ell}^{3}\right) + \left(\frac{2}{15} \cdot \left({\pi}^{5} \cdot {\ell}^{5}\right) + \pi \cdot \ell\right)}}}}{F}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}}\]

    12. Applied simplify1.1

      \[\leadsto \color{blue}{(\left(\frac{-\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\ell \cdot \pi\right)\right) + (\left(\frac{2}{15} \cdot {\pi}^{5}\right) \cdot \left({\ell}^{5}\right) + \left(\ell \cdot \pi\right))_*}} \cdot \left(\sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)} \cdot \sqrt[3]{\tan \left(\ell \cdot \pi\right)}} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right)\right) + \left(\ell \cdot \pi\right))_*}\]

    if 4.004813796983293e-127 < (* PI l)

    1. Initial program 18.1

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

    3. Applied add-sqr-sqrt18.2

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))