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

↓

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

double f(double x_re, double x_im) {
        double r1315429 = x_re;
        double r1315430 = r1315429 * r1315429;
        double r1315431 = x_im;
        double r1315432 = r1315431 * r1315431;
        double r1315433 = r1315430 - r1315432;
        double r1315434 = r1315433 * r1315431;
        double r1315435 = r1315429 * r1315431;
        double r1315436 = r1315431 * r1315429;
        double r1315437 = r1315435 + r1315436;
        double r1315438 = r1315437 * r1315429;
        double r1315439 = r1315434 + r1315438;
        return r1315439;
}

↓

double f(double x_re, double x_im) {
        double r1315440 = x_re;
        double r1315441 = x_im;
        double r1315442 = r1315440 + r1315441;
        double r1315443 = r1315440 - r1315441;
        double r1315444 = r1315442 * r1315443;
        double r1315445 = r1315444 * r1315441;
        double r1315446 = r1315440 * r1315441;
        double r1315447 = r1315441 * r1315440;
        double r1315448 = r1315446 + r1315447;
        double r1315449 = r1315448 * r1315440;
        double r1315450 = r1315445 + r1315449;
        return r1315450;
}

\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 +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