Average Error: 0.0 → 0.0
Time: 5.4s
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 r6731712 = x_re;
        double r6731713 = y_im;
        double r6731714 = r6731712 * r6731713;
        double r6731715 = x_im;
        double r6731716 = y_re;
        double r6731717 = r6731715 * r6731716;
        double r6731718 = r6731714 + r6731717;
        return r6731718;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r6731719 = x_re;
        double r6731720 = y_im;
        double r6731721 = x_im;
        double r6731722 = y_re;
        double r6731723 = r6731721 * r6731722;
        double r6731724 = fma(r6731719, r6731720, r6731723);
        return r6731724;
}

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 2019173 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))