Average Error: 0.0 → 0.0
Time: 1.2s
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 r109660 = x_re;
        double r109661 = y_im;
        double r109662 = r109660 * r109661;
        double r109663 = x_im;
        double r109664 = y_re;
        double r109665 = r109663 * r109664;
        double r109666 = r109662 + r109665;
        return r109666;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r109667 = x_re;
        double r109668 = y_im;
        double r109669 = x_im;
        double r109670 = y_re;
        double r109671 = r109669 * r109670;
        double r109672 = fma(r109667, r109668, r109671);
        return r109672;
}

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 2019326 +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)))