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 -inf 6.9

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

3. Simplified0.2

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

5. Applied add-cube-cbrt0.6

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

6. Applied add-cube-cbrt0.7

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

7. Applied prod-diff0.7

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

8. Applied distribute-lft-in0.7

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

9. Applied associate-+l+0.7

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

10. Simplified0.3

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

11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(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)))