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

↓

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

Error

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

Derivation

Initial program 17.3
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Applied simplify17.3
\[\leadsto \color{blue}{(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(e^{\ell} - e^{-\ell}\right)\right) + U)_*}\]
Taylor expanded around 0 0.4
\[\leadsto (J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3} + \left(2 \cdot \ell + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) + U)_*\]
Applied simplify0.4
\[\leadsto \color{blue}{(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot (\ell \cdot \left((\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + 2)_*\right) + \left(\frac{1}{60} \cdot {\ell}^{5}\right))_*\right) + U)_*}\]

Runtime

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))