Average Error: 16.2 → 8.2
Time: 1.0m
Precision: 64
Internal Precision: 2624
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{1}{\frac{F}{\pi \cdot \ell} - \left(F \cdot \left(\pi \cdot \ell\right)\right) \cdot \frac{1}{3}} \cdot \frac{1}{F}\]

Error

Bits error versus F

Bits error versus l

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Results

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Derivation

  1. Initial program 16.2

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

  2. Initial simplification15.9

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

  4. Applied associate-/r*12.1

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

  6. Applied div-inv12.1

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

  8. Applied *-un-lft-identity12.1

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

  9. Applied associate-/l*12.2

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

  10. Taylor expanded around 0 8.2

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

  11. Final simplification8.2

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))