Average Error: 16.6 → 12.6
Time: 1.3m
Precision: 64
Internal Precision: 2944
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F \cdot \cos \left(\pi \cdot \ell\right)}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 16.6

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

  3. Applied associate-/r*16.6

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

  5. Applied tan-quot16.6

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

  6. Applied frac-times12.7

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

  7. Applied simplify12.6

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))