Average Error: 8.8 → 0.7
Time: 56.8s
Precision: 64
Internal Precision: 128
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}}{F}\]

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 8.8

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

  3. Applied associate-*l/8.3

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

  4. Simplified8.3

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

  6. Applied associate-/r*0.7

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

  8. Applied *-un-lft-identity0.7

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

  9. Applied associate-/l*0.7

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

  10. Final simplification0.7

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

Runtime

Time bar (total: 56.8s)Debug logProfile

herbie shell --seed 2018349 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))