\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

↓

\[(\left((\ell \cdot \left(\sqrt[3]{{\left((\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + 2)_*\right)}^{3}}\right) + \left({\ell}^{5} \cdot \frac{1}{60}\right))_*\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right) + U)_*\]

Derivation

Initial program 17.4
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Taylor expanded around 0 0.4
\[\leadsto \left(J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3} + \left(\frac{1}{60} \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Applied simplify0.4
\[\leadsto \color{blue}{(\left((\ell \cdot \left((\ell \cdot \left(\ell \cdot \frac{1}{3}\right) + 2)_*\right) + \left({\ell}^{5} \cdot \frac{1}{60}\right))_*\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right) + U)_*}\]
Using strategy rm
Applied add-cbrt-cube0.4
\[\leadsto (\left((\ell \cdot \color{blue}{\left(\sqrt[3]{\left((\ell \cdot \left(\ell \cdot \frac{1}{3}\right) + 2)_* \cdot (\ell \cdot \left(\ell \cdot \frac{1}{3}\right) + 2)_*\right) \cdot (\ell \cdot \left(\ell \cdot \frac{1}{3}\right) + 2)_*}\right)} + \left({\ell}^{5} \cdot \frac{1}{60}\right))_*\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right) + U)_*\]
Applied simplify0.4
\[\leadsto (\left((\ell \cdot \left(\sqrt[3]{\color{blue}{{\left((\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + 2)_*\right)}^{3}}}\right) + \left({\ell}^{5} \cdot \frac{1}{60}\right))_*\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right) + U)_*\]

Runtime

herbie shell --seed 2020178 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

Error

Derivation

Runtime