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

↓

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

double f(double x_re, double x_im, double y_re, double y_im) {
        double r529456 = x_re;
        double r529457 = y_im;
        double r529458 = r529456 * r529457;
        double r529459 = x_im;
        double r529460 = y_re;
        double r529461 = r529459 * r529460;
        double r529462 = r529458 + r529461;
        return r529462;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r529463 = y_re;
        double r529464 = x_im;
        double r529465 = y_im;
        double r529466 = x_re;
        double r529467 = r529465 * r529466;
        double r529468 = fma(r529463, r529464, r529467);
        return r529468;
}

x.re \cdot y.im + x.im \cdot y.re

↓

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

Error

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

Derivation

Initial program 0.0
\[x.re \cdot y.im + x.im \cdot y.re\]
Simplified0.0
\[\leadsto \color{blue}{(x.re \cdot y.im + \left(x.im \cdot y.re\right))_*}\]
Taylor expanded around -inf 0.0
\[\leadsto \color{blue}{y.re \cdot x.im + y.im \cdot x.re}\]
Simplified0.0
\[\leadsto \color{blue}{(y.re \cdot x.im + \left(y.im \cdot x.re\right))_*}\]
Final simplification0.0
\[\leadsto (y.re \cdot x.im + \left(y.im \cdot x.re\right))_*\]

Reproduce

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