Average Error: 16.5 → 0.8
Time: 5.9m
Precision: 64
Internal Precision: 3200
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\ell \cdot \pi - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \log \left(e^{\pi \cdot \ell}\right)\right)}}{F}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 16.5

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

  2. Applied simplify16.2

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

  4. Applied associate-/r*12.6

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

  6. Applied clear-num12.6

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

  7. Taylor expanded around 0 8.2

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

  9. Applied add-log-exp0.8

    \[\leadsto \ell \cdot \pi - \frac{\frac{1}{\frac{F}{\pi \cdot \ell} - \frac{1}{3} \cdot \left(F \cdot \color{blue}{\log \left(e^{\pi \cdot \ell}\right)}\right)}}{F}\]
  10. Removed slow pow expressions.

Runtime

Time bar (total: 5.9m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))