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

↓

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

Error

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

Derivation

Initial program 16.9
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Taylor expanded around 0 0.3
\[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Simplified0.3
\[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right) + 2\right) + {\ell}^{5} \cdot \frac{1}{60}\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Using strategy rm
Applied add-log-exp0.5
\[\leadsto \left(J \cdot \left(\ell \cdot \left(\color{blue}{\log \left(e^{\ell \cdot \left(\frac{1}{3} \cdot \ell\right)}\right)} + 2\right) + {\ell}^{5} \cdot \frac{1}{60}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Final simplification0.5
\[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left({\ell}^{5} \cdot \frac{1}{60} + \left(2 + \log \left(e^{\left(\frac{1}{3} \cdot \ell\right) \cdot \ell}\right)\right) \cdot \ell\right)\right)\]

Reproduce

herbie shell --seed 2019088 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))