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

↓

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

double f(double x_re, double x_im) {
        double r1319223 = x_re;
        double r1319224 = r1319223 * r1319223;
        double r1319225 = x_im;
        double r1319226 = r1319225 * r1319225;
        double r1319227 = r1319224 - r1319226;
        double r1319228 = r1319227 * r1319225;
        double r1319229 = r1319223 * r1319225;
        double r1319230 = r1319225 * r1319223;
        double r1319231 = r1319229 + r1319230;
        double r1319232 = r1319231 * r1319223;
        double r1319233 = r1319228 + r1319232;
        return r1319233;
}

↓

double f(double x_re, double x_im) {
        double r1319234 = x_re;
        double r1319235 = x_im;
        double r1319236 = r1319234 + r1319235;
        double r1319237 = r1319234 - r1319235;
        double r1319238 = r1319236 * r1319237;
        double r1319239 = r1319238 * r1319235;
        double r1319240 = r1319234 * r1319235;
        double r1319241 = r1319235 * r1319234;
        double r1319242 = r1319240 + r1319241;
        double r1319243 = r1319242 * r1319234;
        double r1319244 = r1319239 + r1319243;
        return r1319244;
}

\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

↓

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

Derivation

Initial program 0.4
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
Using strategy rm
Applied difference-of-squares0.4
\[\leadsto \frac{\left(\color{blue}{\left(\left(\frac{x.re}{x.im}\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
Final simplification0.4
\[\leadsto \left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

Reproduce

herbie shell --seed 2019102 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))

Error

Derivation

Reproduce