Average Error: 15.9 → 8.0
Time: 1.7m
Precision: 64
Internal Precision: 3456
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\pi \cdot \ell} - \left(\pi \cdot \ell\right) \cdot \left(\frac{1}{3} \cdot F\right)}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 15.9

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

  2. Applied simplify15.7

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

  4. Applied associate-/r*11.9

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

  6. Applied clear-num11.9

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

  7. Taylor expanded around 0 30.4

    \[\leadsto \ell \cdot \pi - \frac{1}{\color{blue}{\frac{{F}^{2}}{\pi \cdot \ell} - \frac{1}{3} \cdot \left({F}^{2} \cdot \left(\pi \cdot \ell\right)\right)}}\]

  8. Applied simplify8.0

    \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\pi \cdot \ell} - \left(\pi \cdot \ell\right) \cdot \left(\frac{1}{3} \cdot F\right)}}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))