Average Error: 16.7 → 14.1
Time: 1.8m
Precision: 64
Internal Precision: 3136
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\pi \cdot \ell - \left(\frac{\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F} \cdot \frac{\sqrt[3]{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F}\right) \cdot \frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)}}{\sqrt[3]{\cos \left(\pi \cdot \ell\right)}}\]

Error

Bits error versus F

Bits error versus l

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 16.7

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

  3. Applied add-cube-cbrt16.9

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

  4. Applied associate-*r*16.9

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

  5. Applied simplify14.1

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

  7. Applied tan-quot14.1

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

  8. Applied cbrt-div14.1

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

  10. Applied tan-quot14.1

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

  12. Applied tan-quot14.1

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2019053 +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))