Average Error: 16.2 → 12.4
Time: 2.0m
Precision: 64
Internal Precision: 3392
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[(\left(-\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}} \cdot \sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}}}\right) \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) + \left(\ell \cdot \pi\right))_*\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 16.2

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

  3. Applied add-cube-cbrt16.3

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

  4. Applied associate-*r*16.3

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

  5. Applied simplify13.6

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

  7. Applied log1p-expm1-u13.8

    \[\leadsto \pi \cdot \ell - \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \sqrt[3]{\color{blue}{\log_* (1 + (e^{\tan \left(\pi \cdot \ell\right)} - 1)^*)}}\]

  8. Taylor expanded around inf 38.3

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

  9. Applied simplify12.3

    \[\leadsto \color{blue}{(\left(-\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) + \left(\ell \cdot \pi\right))_*}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt12.3

    \[\leadsto (\left(-\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}} \cdot \sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}}\right) \cdot \sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}}}} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) + \left(\ell \cdot \pi\right))_*\]

  12. Applied cbrt-prod12.4

    \[\leadsto (\left(-\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}} \cdot \sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\sin \left(\ell \cdot \pi\right)}{\cos \left(\ell \cdot \pi\right)}}}\right)} \cdot \frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) + \left(\ell \cdot \pi\right))_*\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))