Average Error: 0.0 → 0.0
Time: 4.4s
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 r26164 = x_re;
        double r26165 = y_im;
        double r26166 = r26164 * r26165;
        double r26167 = x_im;
        double r26168 = y_re;
        double r26169 = r26167 * r26168;
        double r26170 = r26166 + r26169;
        return r26170;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r26171 = x_re;
        double r26172 = y_im;
        double r26173 = x_im;
        double r26174 = y_re;
        double r26175 = r26173 * r26174;
        double r26176 = fma(r26171, r26172, r26175);
        return r26176;
}

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 2019304 +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)))