Average Error: 8.5 → 0.7
Time: 31.2s
Precision: 64
Internal Precision: 128
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \frac{\log_* (1 + (e^{\tan \left(\pi \cdot \ell\right)} - 1)^*) \cdot \frac{1}{F}}{F}\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 8.5

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

  2. Simplified8.0

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

  4. Applied *-un-lft-identity8.0

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

  5. Applied times-frac0.7

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

  7. Applied log1p-expm1-u0.7

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

  9. Applied associate-*r/0.7

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

  10. Final simplification0.7

    \[\leadsto \pi \cdot \ell - \frac{\log_* (1 + (e^{\tan \left(\pi \cdot \ell\right)} - 1)^*) \cdot \frac{1}{F}}{F}\]

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))