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

↓

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

Error

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

Derivation

Initial program 17.6
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Taylor expanded around 0 0.4
\[\leadsto \left(J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3} + \left(2 \cdot \ell + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Using strategy rm
Applied distribute-lft-in0.4
\[\leadsto \color{blue}{\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \left(2 \cdot \ell + \frac{1}{60} \cdot {\ell}^{5}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U\]
Simplified0.4
\[\leadsto \left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + \color{blue}{J \cdot (\left({\ell}^{5}\right) \cdot \frac{1}{60} + \left(\ell \cdot 2\right))_*}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Final simplification0.4
\[\leadsto \left(\left(\frac{1}{3} \cdot {\ell}^{3}\right) \cdot J + J \cdot (\left({\ell}^{5}\right) \cdot \frac{1}{60} + \left(\ell \cdot 2\right))_*\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

Runtime

herbie shell --seed 2018214 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))