\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
Test:
math.cube on complex, real part
Bits:
128 bits
Bits error versus x.re
Bits error versus x.im
Time: 26.5 s
Input Error: 6.7
Output Error: 0.2
Log: ⚲
Profile: 🕒
\((\left(x.re + x.im\right) * \left({x.re}^2\right) + \left((3 * \left(\left(x.im \cdot x.re\right) \cdot \left(-x.im\right)\right) + \left({x.re}^2 \cdot \left(-x.im\right)\right))_*\right))_*\)
  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
    \[\color{red}{\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im} \leadsto \color{blue}{x.re \cdot \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right) - \left(x.im + x.im\right) \cdot x.im\right)}\]
    6.7
  3. Using strategy rm
    6.7
  4. Applied sub-neg to get
    \[x.re \cdot \left(\left(x.re + x.im\right) \cdot \color{red}{\left(x.re - x.im\right)} - \left(x.im + x.im\right) \cdot x.im\right) \leadsto x.re \cdot \left(\left(x.re + x.im\right) \cdot \color{blue}{\left(x.re + \left(-x.im\right)\right)} - \left(x.im + x.im\right) \cdot x.im\right)\]
    6.7
  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. Using strategy rm
    0.2
  11. Applied add-cbrt-cube to get
    \[{x.re}^2 \cdot \left(x.re + x.im\right) + \color{red}{\left(\left(-\left(x.re + x.im\right)\right) - \left(x.im + x.im\right)\right) \cdot \left(x.im \cdot x.re\right)} \leadsto {x.re}^2 \cdot \left(x.re + x.im\right) + \color{blue}{\sqrt[3]{{\left(\left(\left(-\left(x.re + x.im\right)\right) - \left(x.im + x.im\right)\right) \cdot \left(x.im \cdot x.re\right)\right)}^3}}\]
    20.0
  12. Applied taylor to get
    \[{x.re}^2 \cdot \left(x.re + x.im\right) + \sqrt[3]{{\left(\left(\left(-\left(x.re + x.im\right)\right) - \left(x.im + x.im\right)\right) \cdot \left(x.im \cdot x.re\right)\right)}^3} \leadsto {x.re}^2 \cdot \left(x.re + x.im\right) + \sqrt[3]{{\left(-\left(x.im \cdot {x.re}^2 + 3 \cdot \left({x.im}^2 \cdot x.re\right)\right)\right)}^3}\]
    20.9
  13. Taylor expanded around inf to get
    \[{x.re}^2 \cdot \left(x.re + x.im\right) + \sqrt[3]{{\color{red}{\left(-\left(x.im \cdot {x.re}^2 + 3 \cdot \left({x.im}^2 \cdot x.re\right)\right)\right)}}^3} \leadsto {x.re}^2 \cdot \left(x.re + x.im\right) + \sqrt[3]{{\color{blue}{\left(-\left(x.im \cdot {x.re}^2 + 3 \cdot \left({x.im}^2 \cdot x.re\right)\right)\right)}}^3}\]
    20.9
  14. Applied simplify to get
    \[{x.re}^2 \cdot \left(x.re + x.im\right) + \sqrt[3]{{\left(-\left(x.im \cdot {x.re}^2 + 3 \cdot \left({x.im}^2 \cdot x.re\right)\right)\right)}^3} \leadsto (\left(x.re + x.im\right) * \left({x.re}^2\right) + \left((3 * \left(\left(x.im \cdot x.re\right) \cdot \left(-x.im\right)\right) + \left({x.re}^2 \cdot \left(-x.im\right)\right))_*\right))_*\]
    0.2
  15. Applied final simplification
  16. Removed slow pow expressions

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)))