Average Error: 16.3 → 12.3
Time: 1.9m
Precision: 64
Internal Precision: 3392
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - {\left(\frac{1}{F} \cdot \frac{\tan \left(\ell \cdot \pi\right)}{F}\right)}^{1}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 16.3

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

  3. Applied pow116.3

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

  4. Applied pow116.3

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

  5. Applied pow-prod-down16.3

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

  6. Applied simplify16.1

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

  8. Applied *-un-lft-identity16.1

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

  9. Applied times-frac12.3

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))