Average Error: 16.5 → 12.6
Time: 59.2s
Precision: 64
Internal Precision: 2624
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{F} \cdot \tan \left(\pi \cdot \ell\right)}{F}\]

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 16.5

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

  3. Applied pow116.5

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

  4. Applied pow116.5

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

  5. Applied pow-prod-down16.5

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

  6. Simplified16.2

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

  8. Applied *-un-lft-identity16.2

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

  9. Applied times-frac12.6

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

  11. Applied associate-*r/12.6

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

  12. Final simplification12.6

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

Runtime

Time bar (total: 59.2s)Debug logProfile

herbie shell --seed 2018220 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))