Average Error: 16.1 → 12.1
Time: 1.1m
Precision: 64
Internal Precision: 2368
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[(\left(\frac{-\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\frac{\sqrt[3]{\frac{\sqrt[3]{\sin \left(\ell \cdot \pi\right)} \cdot \sqrt[3]{\sin \left(\ell \cdot \pi\right)}}{\sqrt[3]{\cos \left(\ell \cdot \pi\right)} \cdot \sqrt[3]{\cos \left(\ell \cdot \pi\right)}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sin \left(\ell \cdot \pi\right)}}{\sqrt[3]{\cos \left(\ell \cdot \pi\right)}}}}{\frac{F}{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}}\right) + \left(\ell \cdot \pi\right))_*\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 16.1

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

  3. Applied add-cube-cbrt16.3

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

  4. Applied associate-*r*16.3

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

  5. Applied simplify13.4

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

  7. Applied log1p-expm1-u13.4

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

  8. Taylor expanded around inf 13.4

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

  9. Applied simplify12.0

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

  11. Applied add-cube-cbrt12.0

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

  12. Applied add-cube-cbrt12.0

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

  13. Applied times-frac12.0

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

  14. Applied cbrt-prod12.1

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018166 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))