Average Error: 0.0 → 0.0
Time: 4.2s
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 r35322 = x_re;
        double r35323 = y_im;
        double r35324 = r35322 * r35323;
        double r35325 = x_im;
        double r35326 = y_re;
        double r35327 = r35325 * r35326;
        double r35328 = r35324 + r35327;
        return r35328;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r35329 = x_re;
        double r35330 = y_im;
        double r35331 = x_im;
        double r35332 = y_re;
        double r35333 = r35331 * r35332;
        double r35334 = fma(r35329, r35330, r35333);
        return r35334;
}

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 2019303 +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)))