Average Error: 16.2 → 8.3
Time: 9.2m
Precision: 64
Internal Precision: 3456
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\frac{F}{\pi}}{\ell} - \left(\ell \cdot F\right) \cdot \left(\frac{1}{3} \cdot \pi\right)}\]

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*12.4

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

  6. Applied clear-num12.4

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

  7. Taylor expanded around 0 30.3

    \[\leadsto \ell \cdot \pi - \frac{1}{\color{blue}{\frac{{F}^{2}}{\pi \cdot \ell} - \frac{1}{3} \cdot \left({F}^{2} \cdot \left(\pi \cdot \ell\right)\right)}}\]

  8. Applied simplify8.3

    \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\frac{F}{\pi}}{\ell} - \left(\ell \cdot F\right) \cdot \left(\frac{1}{3} \cdot \pi\right)}}\]
  9. Removed slow pow expressions.

Runtime

Time bar (total: 9.2m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))