Average Error: 8.9 → 0.8
Time: 29.7s
Precision: 64
Internal Precision: 128
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\frac{1}{\tan \left(\pi \cdot \ell\right)}}{\frac{1}{F}}}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 8.9

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

  2. Simplified8.5

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

  4. Applied associate-/r*0.7

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

  6. Applied clear-num0.7

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

  8. Applied div-inv0.7

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

  9. Applied associate-/r*0.7

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

  11. Applied div-inv0.8

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

  12. Applied associate-/r*0.8

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

  13. Final simplification0.8

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

Reproduce

herbie shell --seed 2019053 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))