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

↓

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

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2581038 = x_re;
        double r2581039 = y_re;
        double r2581040 = r2581038 * r2581039;
        double r2581041 = x_im;
        double r2581042 = y_im;
        double r2581043 = r2581041 * r2581042;
        double r2581044 = r2581040 - r2581043;
        return r2581044;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2581045 = x_re;
        double r2581046 = y_re;
        double r2581047 = x_im;
        double r2581048 = y_im;
        double r2581049 = r2581047 * r2581048;
        double r2581050 = -r2581049;
        double r2581051 = fma(r2581045, r2581046, r2581050);
        return r2581051;
}

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 2019102 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))