Average Error: 0.0 → 0.0
Time: 3.9s
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 r40614 = x_re;
        double r40615 = y_im;
        double r40616 = r40614 * r40615;
        double r40617 = x_im;
        double r40618 = y_re;
        double r40619 = r40617 * r40618;
        double r40620 = r40616 + r40619;
        return r40620;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r40621 = x_re;
        double r40622 = y_im;
        double r40623 = x_im;
        double r40624 = y_re;
        double r40625 = r40623 * r40624;
        double r40626 = fma(r40621, r40622, r40625);
        return r40626;
}

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 2019325 +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)))