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. Using strategy rm

4. Applied distribute-rgt-in0.4

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

5. Applied associate-+r+0.4

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

7. Applied associate-+l+0.4

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

8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019094 +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)))