\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

↓

\[x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\]

Error

The X axis uses an exponential scale — Bits error versus `x.re`

Derivation

Initial program 0.4
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
Simplified0.4
\[\leadsto \color{blue}{x.im \cdot \left(\frac{\left(\left(\frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)}{\left(x.re \cdot \left(\frac{x.re}{x.re}\right)\right)}\right)}\]
Final simplification0.4
\[\leadsto x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))