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

↓

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

Error

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

Derivation

Initial program 17.3
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Taylor expanded around 0 0.4
\[\leadsto \left(J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3} + \left(2 \cdot \ell + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Taylor expanded around -inf 62.5
\[\leadsto \left(J \cdot \left(\color{blue}{\frac{1}{3} \cdot e^{3 \cdot \left(\log -1 - \log \left(\frac{-1}{\ell}\right)\right)}} + \left(2 \cdot \ell + \frac{1}{60} \cdot {\ell}^{5}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Applied simplify0.4
\[\leadsto \color{blue}{U + \left(\left(\frac{1}{60} \cdot {\ell}^{5} + {\ell}^{3} \cdot \frac{1}{3}\right) + \left(\ell + \ell\right)\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\]

Runtime

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))