Average Error: 0.0 → 0.0
Time: 1.3s
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 r738466 = x_re;
        double r738467 = y_im;
        double r738468 = r738466 * r738467;
        double r738469 = x_im;
        double r738470 = y_re;
        double r738471 = r738469 * r738470;
        double r738472 = r738468 + r738471;
        return r738472;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r738473 = x_re;
        double r738474 = y_im;
        double r738475 = x_im;
        double r738476 = y_re;
        double r738477 = r738475 * r738476;
        double r738478 = fma(r738473, r738474, r738477);
        return r738478;
}

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)))