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

↓

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

Error

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

Derivation

Initial program 17.6
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Initial simplification17.7
\[\leadsto (\left(\cos \left(\frac{K}{2}\right)\right) \cdot \left(J \cdot e^{\ell} - \frac{J}{e^{\ell}}\right) + U)_*\]
Taylor expanded around 0 0.5
\[\leadsto (\left(\cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} + U)_*\]
Final simplification0.5
\[\leadsto (\left(\cos \left(\frac{K}{2}\right)\right) \cdot \left(\left(J \cdot \ell\right) \cdot 2\right) + U)_*\]

Runtime

herbie shell --seed 2018216 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))