Average Error: 8.6 → 0.7
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F} \cdot \frac{1}{F}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 8.6

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

  2. Simplified8.1

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

  4. Applied *-un-lft-identity8.1

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

  5. Applied associate-*r*8.1

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

  6. Applied *-un-lft-identity8.1

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

  7. Applied times-frac0.7

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

  8. Simplified0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2019088 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))