Average Error: 0.0 → 0.0
Time: 3.5s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r89610 = x_re;
        double r89611 = y_im;
        double r89612 = r89610 * r89611;
        double r89613 = x_im;
        double r89614 = y_re;
        double r89615 = r89613 * r89614;
        double r89616 = r89612 + r89615;
        return r89616;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r89617 = x_re;
        double r89618 = y_im;
        double r89619 = x_im;
        double r89620 = y_re;
        double r89621 = r89619 * r89620;
        double r89622 = fma(r89617, r89618, r89621);
        return r89622;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))