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

↓

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

Error

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

Derivation

Initial program 17.0
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
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 add-cube-cbrt0.8
\[\leadsto \color{blue}{\left(\sqrt[3]{\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3} + \left(2 \cdot \ell + \frac{1}{60} \cdot {\ell}^{5}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3} + \left(2 \cdot \ell + \frac{1}{60} \cdot {\ell}^{5}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt[3]{\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3} + \left(2 \cdot \ell + \frac{1}{60} \cdot {\ell}^{5}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)}} + U\]

Runtime

herbie shell --seed 2018207 
(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