\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]

↓

\[(\left(\left(x.re + x.im\right) \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(\left(-2 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right))_*\]

Error

The X axis uses an exponential scale — Bits error versus `x.re`

Derivation

Initial program 6.9
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
Taylor expanded around inf 6.9
\[\leadsto \left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \color{blue}{2 \cdot \left(x.re \cdot {x.im}^{2}\right)}\]
Applied simplify0.2
\[\leadsto \color{blue}{(\left(\left(x.re + x.im\right) \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(\left(-2 \cdot x.im\right) \cdot \left(x.re \cdot x.im\right)\right))_*}\]

Runtime

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (x.re x.im)
  :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)))