Average Error: 15.9 → 12.1
Time: 57.6s
Precision: 64
Internal Precision: 2624
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\]

Error

Bits error versus F

Bits error versus l

Try it out

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 pow115.9

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

  4. Applied pow115.9

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

  5. Applied pow-prod-down15.9

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

  6. Simplified15.7

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

  8. Applied associate-/r*12.1

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

  9. Final simplification12.1

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

Runtime

Time bar (total: 57.6s)Debug logProfile

herbie shell --seed 2018221 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))