Average Error: 15.9 → 12.2
Time: 2.6m
Precision: 64
Internal Precision: 3904
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[(\left(-\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)}}{F}\right) \cdot \left(\frac{\sqrt[3]{\tan \left(\ell \cdot \pi\right)} \cdot \sqrt[3]{\sin \left(\ell \cdot \pi\right)}}{F \cdot \sqrt[3]{\cos \left(\ell \cdot \pi\right)}}\right) + \left(\ell \cdot \pi\right))_*\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 15.9

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

  3. Applied add-cube-cbrt16.1

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

  4. Applied associate-*r*16.1

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

  5. Applied simplify13.4

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

  7. Applied tan-quot13.4

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

  8. Applied cbrt-div13.4

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

  9. Taylor expanded around inf 37.5

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

  10. Applied simplify12.2

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))