\[x.re \cdot y.re - x.im \cdot y.im\]

↓

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

double f(double x_re, double x_im, double y_re, double y_im) {
        double r527804 = x_re;
        double r527805 = y_re;
        double r527806 = r527804 * r527805;
        double r527807 = x_im;
        double r527808 = y_im;
        double r527809 = r527807 * r527808;
        double r527810 = r527806 - r527809;
        return r527810;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r527811 = x_re;
        double r527812 = y_re;
        double r527813 = x_im;
        double r527814 = y_im;
        double r527815 = r527813 * r527814;
        double r527816 = -r527815;
        double r527817 = fma(r527811, r527812, r527816);
        return r527817;
}

x.re \cdot y.re - x.im \cdot y.im

↓

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

Error

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

Derivation

Initial program 0.0
\[x.re \cdot y.re - x.im \cdot y.im\]
Using strategy rm
Applied fma-neg0.0
\[\leadsto \color{blue}{(x.re \cdot y.re + \left(-x.im \cdot y.im\right))_*}\]
Final simplification0.0
\[\leadsto (x.re \cdot y.re + \left(-x.im \cdot y.im\right))_*\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))