Average Error: 16.2 → 8.4
Time: 1.6m
Precision: 64
Internal Precision: 3392
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{(\frac{-1}{3} \cdot \left(\left(\pi \cdot \ell\right) \cdot F\right) + \left(\frac{F}{\pi \cdot \ell}\right))_*}}{F}\]

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. Using strategy rm

  3. Applied associate-*l/16.0

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

  4. Simplified16.0

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

  6. Applied associate-/r*12.4

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

  8. Applied *-un-lft-identity12.4

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

  9. Applied associate-/l*12.4

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

  10. Taylor expanded around 0 8.4

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

  11. Simplified8.4

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

  12. Final simplification8.4

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{(\frac{-1}{3} \cdot \left(\left(\pi \cdot \ell\right) \cdot F\right) + \left(\frac{F}{\pi \cdot \ell}\right))_*}}{F}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018256 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))