Average Error: 15.7 → 11.8
Time: 1.4m
Precision: 64
Internal Precision: 2880
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\frac{\frac{\sin \left(\left(\ell \cdot \pi\right) \cdot \sqrt[3]{-1}\right)}{F}}{F \cdot \cos \left(\left(\ell \cdot \pi\right) \cdot \sqrt[3]{-1}\right)} + \ell \cdot \pi\]

Error

Bits error versus F

Bits error versus l

Derivation

  1. Initial program 15.7

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

  3. Applied add-cube-cbrt15.8

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

  4. Taylor expanded around -inf 62.4

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\sin \left(-1 \cdot \left(\left(\pi \cdot \ell\right) \cdot {-1}^{\frac{1}{3}}\right)\right)}{\cos \left(-1 \cdot \left(\left(\pi \cdot \ell\right) \cdot {-1}^{\frac{1}{3}}\right)\right) \cdot {F}^{2}}}\]

  5. Applied simplify11.8

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' +o rules:numerics
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  (- (* PI l) (* (/ 1 (* F F)) (tan (* PI l)))))