\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
Test:
math.cube on complex, imaginary part
Bits:
128 bits
Bits error versus x.re
Bits error versus x.im
Time: 10.3 s
Input Error: 3.2
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right) - {x.im}^3\)
  1. Started with
    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
    3.2
  2. Applied simplify to get
    \[\color{red}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re} \leadsto \color{blue}{x.im \cdot \left(\left(x.re + x.re\right) \cdot x.re + \left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)}\]
    3.3
  3. Applied taylor to get
    \[x.im \cdot \left(\left(x.re + x.re\right) \cdot x.re + \left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \leadsto 3 \cdot \left(x.im \cdot {x.re}^2\right) - {x.im}^{3}\]
    3.2
  4. Taylor expanded around inf to get
    \[\color{red}{3 \cdot \left(x.im \cdot {x.re}^2\right) - {x.im}^{3}} \leadsto \color{blue}{3 \cdot \left(x.im \cdot {x.re}^2\right) - {x.im}^{3}}\]
    3.2
  5. Using strategy rm
    3.2
  6. Applied square-mult to get
    \[3 \cdot \left(x.im \cdot \color{red}{{x.re}^2}\right) - {x.im}^{3} \leadsto 3 \cdot \left(x.im \cdot \color{blue}{\left(x.re \cdot x.re\right)}\right) - {x.im}^{3}\]
    3.2
  7. Applied associate-*r* to get
    \[3 \cdot \color{red}{\left(x.im \cdot \left(x.re \cdot x.re\right)\right)} - {x.im}^{3} \leadsto 3 \cdot \color{blue}{\left(\left(x.im \cdot x.re\right) \cdot x.re\right)} - {x.im}^{3}\]
    0.2
  8. Applied taylor to get
    \[3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - {x.im}^{3} \leadsto 3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - {x.im}^{3}\]
    0.2
  9. Taylor expanded around 0 to get
    \[3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - \color{red}{{x.im}^{3}} \leadsto 3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - \color{blue}{{x.im}^{3}}\]
    0.2
  10. Applied simplify to get
    \[3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - {x.im}^{3} \leadsto \left(x.re \cdot 3\right) \cdot \left(x.im \cdot x.re\right) - {x.im}^3\]
    0.3
  11. Applied final simplification

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)))