Average Error: 15.9 → 12.0
Time: 1.2m
Precision: 64
Internal Precision: 2880
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F}}{F}\]

Error

Bits error versus F

Bits error versus l

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.9

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

  3. Applied associate-*l/15.6

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

  4. Simplified15.6

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

  6. Applied associate-/r*12.0

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

  8. Applied div-inv12.0

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

  9. Taylor expanded around inf 12.0

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

  10. Final simplification12.0

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

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)))))