Error

Derivation

1. Initial program 0.4

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

2. Simplified0.4

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

3. Simplified0.4

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

5. Applied p16-distribute-lft-out0.4

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

6. Applied p16-distribute-lft-out0.4

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

7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019089 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (-.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.re) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.im)))