Error

Target

Derivation

1. Initial program 7.0

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

2. Taylor expanded around 0 6.9

   \[\leadsto \color{blue}{\left({x.re}^{3} - {x.im}^{2} \cdot x.re\right)} - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]

3. Simplified0.2

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

5. Applied fma-neg0.2

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

6. Final simplification0.2

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

Reproduce

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

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))