Average Error: 8.4 → 1.1
Time: 40.0s
Precision: 64
Internal Precision: 128
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\sqrt[3]{\tan \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\tan \left(\pi \cdot \ell\right)}}{\frac{F}{\sqrt[3]{\tan \left(\pi \cdot \ell\right)}}}}{F}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 8.4

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

  2. Simplified7.9

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

  4. Applied associate-/r*0.7

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

  6. Applied add-cube-cbrt1.1

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

  7. Applied associate-/l*1.1

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

  8. Final simplification1.1

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

Reproduce

herbie shell --seed 2019094 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))