\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
Test:
math.cube on complex, imaginary part
Bits:
128 bits
Bits error versus x.re
Bits error versus x.im
Time: 21.2 s
Input Error: 3.3
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\left(\left(x.re + x.re\right) + x.re\right) \cdot \left(x.im \cdot x.re\right) - {x.im}^3\)
  1. Started with
    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
    3.3
  2. Applied simplify to get
    \[\color{red}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re} \leadsto \color{blue}{(\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) * x.im + \left(\left(x.im + x.im\right) \cdot {x.re}^2\right))_*}\]
    3.3
  3. Using strategy rm
    3.3
  4. Applied fma-udef to get
    \[\color{red}{(\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) * x.im + \left(\left(x.im + x.im\right) \cdot {x.re}^2\right))_*} \leadsto \color{blue}{\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(x.im + x.im\right) \cdot {x.re}^2}\]
    3.3
  5. Applied taylor to get
    \[\left(\left(x.re - x.im\right) \cdot \left(x.re + x.im\right)\right) \cdot x.im + \left(x.im + x.im\right) \cdot {x.re}^2 \leadsto \left(x.im \cdot {x.re}^2 - {x.im}^{3}\right) + \left(x.im + x.im\right) \cdot {x.re}^2\]
    3.3
  6. Taylor expanded around 0 to get
    \[\color{red}{\left(x.im \cdot {x.re}^2 - {x.im}^{3}\right)} + \left(x.im + x.im\right) \cdot {x.re}^2 \leadsto \color{blue}{\left(x.im \cdot {x.re}^2 - {x.im}^{3}\right)} + \left(x.im + x.im\right) \cdot {x.re}^2\]
    3.3
  7. Applied taylor to get
    \[\left(x.im \cdot {x.re}^2 - {x.im}^{3}\right) + \left(x.im + x.im\right) \cdot {x.re}^2 \leadsto \left(x.im \cdot {x.re}^2 - {x.im}^{-3}\right) + \left(x.im + x.im\right) \cdot {x.re}^2\]
    28.9
  8. Taylor expanded around inf to get
    \[\color{red}{\left(x.im \cdot {x.re}^2 - {x.im}^{-3}\right)} + \left(x.im + x.im\right) \cdot {x.re}^2 \leadsto \color{blue}{\left(x.im \cdot {x.re}^2 - {x.im}^{-3}\right)} + \left(x.im + x.im\right) \cdot {x.re}^2\]
    28.9
  9. Applied simplify to get
    \[\color{red}{\left(x.im \cdot {x.re}^2 - {x.im}^{-3}\right) + \left(x.im + x.im\right) \cdot {x.re}^2} \leadsto \color{blue}{\left(x.re \cdot x.im\right) \cdot \left(\left(x.re + x.re\right) + x.re\right) - {x.im}^{-3}}\]
    28.9
  10. Applied taylor to get
    \[\left(x.re \cdot x.im\right) \cdot \left(\left(x.re + x.re\right) + x.re\right) - {x.im}^{-3} \leadsto \left(x.re \cdot x.im\right) \cdot \left(\left(x.re + x.re\right) + x.re\right) - {x.im}^{3}\]
    0.2
  11. Taylor expanded around inf to get
    \[\left(x.re \cdot x.im\right) \cdot \left(\left(x.re + x.re\right) + x.re\right) - \color{red}{{x.im}^{3}} \leadsto \left(x.re \cdot x.im\right) \cdot \left(\left(x.re + x.re\right) + x.re\right) - \color{blue}{{x.im}^{3}}\]
    0.2
  12. Applied simplify to get
    \[\left(x.re \cdot x.im\right) \cdot \left(\left(x.re + x.re\right) + x.re\right) - {x.im}^{3} \leadsto \left(\left(x.re + x.re\right) + x.re\right) \cdot \left(x.im \cdot x.re\right) - {x.im}^3\]
    0.3
  13. Applied final simplification
  14. Removed slow pow expressions

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)))