Average Error: 16.0 → 12.6
Time: 1.2m
Precision: 64
Internal Precision: 2432
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\ell \cdot \pi - \left(\left(\sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right)\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 16.0

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

  3. Applied add-cube-cbrt16.2

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

  4. Applied associate-*r*16.2

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

  5. Applied simplify13.6

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

  7. Applied add-cube-cbrt13.6

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

  9. Applied add-cube-cbrt13.7

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

  10. Taylor expanded around inf 13.7

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

  11. Applied simplify12.6

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))