Error

Target

Derivation

1. Initial program 6.9

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

2. Taylor expanded around 0 6.8

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

3. Simplified0.2

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

5. Applied add-cube-cbrt0.6

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

6. Applied associate-*l*0.6

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

7. Final simplification0.6

   \[\leadsto \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right) + \left(\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}\right) \cdot \left(\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot x.re\right)\]

Reproduce

herbie shell --seed 2019068 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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