Average Error: 0.0 → 0.0
Time: 1.3s
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 r26740 = x_re;
        double r26741 = y_im;
        double r26742 = r26740 * r26741;
        double r26743 = x_im;
        double r26744 = y_re;
        double r26745 = r26743 * r26744;
        double r26746 = r26742 + r26745;
        return r26746;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r26747 = x_re;
        double r26748 = y_im;
        double r26749 = x_im;
        double r26750 = y_re;
        double r26751 = r26749 * r26750;
        double r26752 = fma(r26747, r26748, r26751);
        return r26752;
}

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 2019323 +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)))