Average Error: 0.0 → 0.0
Time: 3.4s
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 r2130624 = x_re;
        double r2130625 = y_im;
        double r2130626 = r2130624 * r2130625;
        double r2130627 = x_im;
        double r2130628 = y_re;
        double r2130629 = r2130627 * r2130628;
        double r2130630 = r2130626 + r2130629;
        return r2130630;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2130631 = x_re;
        double r2130632 = y_im;
        double r2130633 = x_im;
        double r2130634 = y_re;
        double r2130635 = r2130633 * r2130634;
        double r2130636 = fma(r2130631, r2130632, r2130635);
        return r2130636;
}

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 2019134 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))