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

↓

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

double f(double x_re, double x_im) {
        double r312150 = x_re;
        double r312151 = r312150 * r312150;
        double r312152 = x_im;
        double r312153 = r312152 * r312152;
        double r312154 = r312151 - r312153;
        double r312155 = r312154 * r312152;
        double r312156 = r312150 * r312152;
        double r312157 = r312152 * r312150;
        double r312158 = r312156 + r312157;
        double r312159 = r312158 * r312150;
        double r312160 = r312155 + r312159;
        return r312160;
}

↓

double f(double x_re, double x_im) {
        double r312161 = x_im;
        double r312162 = x_re;
        double r312163 = r312161 + r312162;
        double r312164 = r312162 - r312161;
        double r312165 = r312163 * r312164;
        double r312166 = r312162 + r312162;
        double r312167 = r312162 * r312166;
        double r312168 = r312165 + r312167;
        double r312169 = r312161 * r312168;
        return r312169;
}

\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

↓

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

Derivation

Initial program 0.4
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
Simplified0.4
\[\leadsto \color{blue}{x.im \cdot \left(\frac{\left(\left(\frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)}{\left(x.re \cdot \left(\frac{x.re}{x.re}\right)\right)}\right)}\]
Using strategy rm
Applied distribute-rgt-in0.4
\[\leadsto \color{blue}{\frac{\left(\left(\left(\frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right) \cdot x.im\right)}{\left(\left(x.re \cdot \left(\frac{x.re}{x.re}\right)\right) \cdot x.im\right)}}\]
Using strategy rm
Applied distribute-rgt-out0.4
\[\leadsto \color{blue}{x.im \cdot \left(\frac{\left(\left(\frac{x.im}{x.re}\right) \cdot \left(x.re - x.im\right)\right)}{\left(x.re \cdot \left(\frac{x.re}{x.re}\right)\right)}\right)}\]
Final simplification0.4
\[\leadsto x.im \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re + x.re\right)\right)\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(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