Average Error: 16.5 → 16.7
Time: 2.4m
Precision: 64
Internal Precision: 3904
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[(\left(\frac{\sqrt[3]{\left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right) \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}}}{\sqrt[3]{\cos \left(\pi \cdot \ell\right)}} \cdot \left(\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\right)\right) \cdot \left(\frac{-1}{F \cdot F}\right) + \left(\pi \cdot \ell\right))_*\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 16.5

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

  2. Initial simplification16.5

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

  4. Applied add-cube-cbrt16.7

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

  6. Applied tan-quot16.7

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

  7. Applied cbrt-div16.7

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

  9. Applied add-cube-cbrt16.7

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

  10. Final simplification16.7

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))