\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
Test:
math.cube on complex, imaginary part
Bits:
128 bits
Bits error versus x.re
Bits error versus x.im
Time: 5.8 s
Input Error: 6.9
Output Error: 6.9
Log: ⚲
Profile: 🕒
\((\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) * x.im + \left(\left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.re\right))_*\)
  1. Started with
    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
    6.9
  2. Applied simplify to get
    \[\color{red}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re} \leadsto \color{blue}{(\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) * x.im + \left(\left(x.im + x.im\right) \cdot {x.re}^2\right))_*}\]
    6.9
  3. Using strategy rm
    6.9
  4. Applied square-mult to get
    \[(\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) * x.im + \left(\left(x.im + x.im\right) \cdot \color{red}{{x.re}^2}\right))_* \leadsto (\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) * x.im + \left(\left(x.im + x.im\right) \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right))_*\]
    6.9
  5. Applied associate-*r* to get
    \[(\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) * x.im + \color{red}{\left(\left(x.im + x.im\right) \cdot \left(x.re \cdot x.re\right)\right)})_* \leadsto (\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) * x.im + \color{blue}{\left(\left(\left(x.im + x.im\right) \cdot x.re\right) \cdot x.re\right)})_*\]
    6.9

Original test:


(lambda ((x.re default) (x.im default))
  #:name "math.cube on complex, imaginary part"
  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))