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

↓

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

Error

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

Derivation

Initial program 17.2
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Taylor expanded around 0 0.4
\[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
Using strategy rm
Applied associate-*l*0.4
\[\leadsto \color{blue}{J \cdot \left(\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U\]
Final simplification0.4
\[\leadsto U + J \cdot \left(\left(\ell \cdot 2 + \left(\frac{1}{60} \cdot {\ell}^{5} + {\ell}^{3} \cdot \frac{1}{3}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.4	0.4	0.1	0.3	0%

herbie shell --seed 2018290 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))

In
Out

Error

Try it out

Derivation

Runtime