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

↓

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

Error

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

Derivation

Initial program 17.3
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Taylor expanded around 0 0.5
\[\leadsto \left(J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3} + \left(2 \cdot \ell + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Applied simplify0.5
\[\leadsto \color{blue}{(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left((\ell \cdot \left((\left(\ell \cdot \frac{1}{3}\right) \cdot \ell + 2)_*\right) + \left({\ell}^{5} \cdot \frac{1}{60}\right))_*\right) + U)_*}\]
Using strategy rm
Applied fma-udef0.5
\[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot (\ell \cdot \left((\left(\ell \cdot \frac{1}{3}\right) \cdot \ell + 2)_*\right) + \left({\ell}^{5} \cdot \frac{1}{60}\right))_* + U}\]

Runtime

herbie shell --seed 2018170 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))