\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
Test:
math.cube on complex, real part
Bits:
128 bits
Bits error versus x.re
Bits error versus x.im
Time: 27.7 s
Input Error: 6.7
Output Error: 0.2
Log: ⚲
Profile: 🕒
\((x.im * \left(\left(-x.re\right) \cdot (3 * x.im + x.re)_*\right) + \left(\left(x.re + x.im\right) \cdot {x.re}^2\right))_*\)
  1. Started with
    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
    6.7
  2. Applied simplify to get
    \[\color{red}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im} \leadsto \color{blue}{x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)}\]
    6.8
  3. Using strategy rm
    6.8
  4. Applied sub-neg to get
    \[x.re \cdot \left(\left(x.re + x.im\right) \cdot \color{red}{\left(x.re - x.im\right)} - \left(x.im + x.im\right) \cdot x.im\right) \leadsto x.re \cdot \left(\left(x.re + x.im\right) \cdot \color{blue}{\left(x.re + \left(-x.im\right)\right)} - \left(x.im + x.im\right) \cdot x.im\right)\]
    6.8
  5. Applied distribute-lft-in to get
    \[x.re \cdot \left(\color{red}{\left(x.re + x.im\right) \cdot \left(x.re + \left(-x.im\right)\right)} - \left(x.im + x.im\right) \cdot x.im\right) \leadsto x.re \cdot \left(\color{blue}{\left(\left(x.re + x.im\right) \cdot x.re + \left(x.re + x.im\right) \cdot \left(-x.im\right)\right)} - \left(x.im + x.im\right) \cdot x.im\right)\]
    6.8
  6. Applied associate--l+ to get
    \[x.re \cdot \color{red}{\left(\left(\left(x.re + x.im\right) \cdot x.re + \left(x.re + x.im\right) \cdot \left(-x.im\right)\right) - \left(x.im + x.im\right) \cdot x.im\right)} \leadsto x.re \cdot \color{blue}{\left(\left(x.re + x.im\right) \cdot x.re + \left(\left(x.re + x.im\right) \cdot \left(-x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)\right)}\]
    6.8
  7. Applied distribute-lft-in to get
    \[\color{red}{x.re \cdot \left(\left(x.re + x.im\right) \cdot x.re + \left(\left(x.re + x.im\right) \cdot \left(-x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)\right)} \leadsto \color{blue}{x.re \cdot \left(\left(x.re + x.im\right) \cdot x.re\right) + x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(-x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)}\]
    6.8
  8. Applied simplify to get
    \[\color{red}{x.re \cdot \left(\left(x.re + x.im\right) \cdot x.re\right)} + x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(-x.im\right) - \left(x.im + x.im\right) \cdot x.im\right) \leadsto \color{blue}{{x.re}^2 \cdot \left(x.re + x.im\right)} + x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(-x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)\]
    6.8
  9. Applied simplify to get
    \[{x.re}^2 \cdot \left(x.re + x.im\right) + \color{red}{x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(-x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)} \leadsto {x.re}^2 \cdot \left(x.re + x.im\right) + \color{blue}{\left(\left(-\left(x.re + x.im\right)\right) - \left(x.im + x.im\right)\right) \cdot \left(x.im \cdot x.re\right)}\]
    0.2
  10. Applied taylor to get
    \[{x.re}^2 \cdot \left(x.re + x.im\right) + \left(\left(-\left(x.re + x.im\right)\right) - \left(x.im + x.im\right)\right) \cdot \left(x.im \cdot x.re\right) \leadsto {x.re}^2 \cdot \left(x.re + x.im\right) + \left(-\left(3 \cdot x.im + x.re\right)\right) \cdot \left(x.im \cdot x.re\right)\]
    0.2
  11. Taylor expanded around 0 to get
    \[{x.re}^2 \cdot \left(x.re + x.im\right) + \color{red}{\left(-\left(3 \cdot x.im + x.re\right)\right)} \cdot \left(x.im \cdot x.re\right) \leadsto {x.re}^2 \cdot \left(x.re + x.im\right) + \color{blue}{\left(-\left(3 \cdot x.im + x.re\right)\right)} \cdot \left(x.im \cdot x.re\right)\]
    0.2
  12. Applied simplify to get
    \[\color{red}{{x.re}^2 \cdot \left(x.re + x.im\right) + \left(-\left(3 \cdot x.im + x.re\right)\right) \cdot \left(x.im \cdot x.re\right)} \leadsto \color{blue}{(x.im * \left(\left(-x.re\right) \cdot (3 * x.im + x.re)_*\right) + \left(\left(x.re + x.im\right) \cdot {x.re}^2\right))_*}\]
    0.2

Original test:


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