Error

Derivation

1. Initial program 0.4

   \[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]

2. Simplified0.4

   \[\leadsto \color{blue}{x.im \cdot \left(\frac{\left(\left(\frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)}{\left(x.re \cdot \left(\frac{x.re}{x.re}\right)\right)}\right)}\]

3. Simplified0.4

   \[\leadsto \color{blue}{\left(\left(x.re \cdot \left(\frac{\left(\frac{x.re}{x.re}\right)}{x.re}\right)\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im}\]
4. Using strategy rm

5. Applied distribute-rgt-in0.4

   \[\leadsto \left(\color{blue}{\left(\frac{\left(\left(\frac{x.re}{x.re}\right) \cdot x.re\right)}{\left(x.re \cdot x.re\right)}\right)} - \left(x.im \cdot x.im\right)\right) \cdot x.im\]

6. Applied associate--l+0.4

   \[\leadsto \color{blue}{\left(\frac{\left(\left(\frac{x.re}{x.re}\right) \cdot x.re\right)}{\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right)}\right)} \cdot x.im\]

7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019100 +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)))