Average Error: 0.0 → 0.0
Time: 1.1s
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 r672796 = x_re;
        double r672797 = y_im;
        double r672798 = r672796 * r672797;
        double r672799 = x_im;
        double r672800 = y_re;
        double r672801 = r672799 * r672800;
        double r672802 = r672798 + r672801;
        return r672802;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r672803 = x_re;
        double r672804 = y_im;
        double r672805 = x_im;
        double r672806 = y_re;
        double r672807 = r672805 * r672806;
        double r672808 = fma(r672803, r672804, r672807);
        return r672808;
}

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 2019153 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))