Average Error: 15.9 → 0.8
Time: 1.2m
Precision: 64
Internal Precision: 2880
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{(\left(F \cdot \frac{-1}{3}\right) \cdot \left(\log \left(e^{\pi \cdot \ell}\right)\right) + \left(\frac{F}{\pi \cdot \ell}\right))_*}}{F}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 15.9

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

  2. Initial simplification12.0

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

  4. Applied clear-num12.0

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

  5. Taylor expanded around 0 8.1

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

  6. Simplified8.1

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

  8. Applied add-log-exp0.8

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

  9. Final simplification0.8

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))