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. Using strategy rm

3. Applied *-commutative0.4

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

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

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

5. Applied associate-*l*0.4

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

6. Applied *-commutative0.4

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

7. Applied distribute-lft-out--0.4

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

8. Simplified0.4

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

9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019072 
(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)))