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

↓

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

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1794051 = x_re;
        double r1794052 = y_im;
        double r1794053 = r1794051 * r1794052;
        double r1794054 = x_im;
        double r1794055 = y_re;
        double r1794056 = r1794054 * r1794055;
        double r1794057 = r1794053 + r1794056;
        return r1794057;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1794058 = y_re;
        double r1794059 = x_im;
        double r1794060 = y_im;
        double r1794061 = x_re;
        double r1794062 = r1794060 * r1794061;
        double r1794063 = fma(r1794058, r1794059, r1794062);
        return r1794063;
}

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