Average Error: 16.0 → 13.5
Time: 1.8m
Precision: 64
Internal Precision: 3392
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}\]

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-cbrt16.1

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

  4. Applied associate-*r*16.1

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

  5. Applied simplify13.5

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))