Average Error: 16.2 → 12.5
Time: 2.0m
Precision: 64
Internal Precision: 3712
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\ell \cdot \pi - \left(\sqrt[3]{\frac{\frac{\tan \left(\ell \cdot \pi\right)}{F}}{F}} \cdot \sqrt[3]{\frac{\frac{\tan \left(\ell \cdot \pi\right)}{F}}{F}}\right) \cdot \frac{\sqrt[3]{\frac{\tan \left(\ell \cdot \pi\right)}{F}}}{\sqrt[3]{F}}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 16.2

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

  2. Applied simplify16.0

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

  4. Applied associate-/r*12.3

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

  6. Applied add-cube-cbrt12.5

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

  8. Applied cbrt-div12.5

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))