Average Error: 0.0 → 0.0
Time: 1.6s
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 r29713 = x_re;
        double r29714 = y_im;
        double r29715 = r29713 * r29714;
        double r29716 = x_im;
        double r29717 = y_re;
        double r29718 = r29716 * r29717;
        double r29719 = r29715 + r29718;
        return r29719;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r29720 = x_re;
        double r29721 = y_im;
        double r29722 = x_im;
        double r29723 = y_re;
        double r29724 = r29722 * r29723;
        double r29725 = fma(r29720, r29721, r29724);
        return r29725;
}

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)))