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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `J`

Derivation

Initial program 16.8
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Taylor expanded around 0 0.4
\[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Applied simplify0.4
\[\leadsto \color{blue}{(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left((\ell \cdot \left((\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + 2)_*\right) + \left(\frac{1}{60} \cdot {\ell}^{5}\right))_*\right) + U)_*}\]
Taylor expanded around inf 0.4
\[\leadsto \color{blue}{2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \ell\right)\right) + \left(\frac{1}{3} \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot {\ell}^{3}\right)\right) + U\right)}\]
Applied simplify0.4
\[\leadsto \color{blue}{(J \cdot \left(\cos \left(K \cdot \frac{1}{2}\right) \cdot (\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} \cdot \ell\right) + \left(2 \cdot \ell\right))_*\right) + U)_*}\]

Runtime

herbie shell --seed 2019053 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))