Average Error: 0.0 → 0.0
Time: 5.9s
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 r23533 = x_re;
        double r23534 = y_im;
        double r23535 = r23533 * r23534;
        double r23536 = x_im;
        double r23537 = y_re;
        double r23538 = r23536 * r23537;
        double r23539 = r23535 + r23538;
        return r23539;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r23540 = x_re;
        double r23541 = y_im;
        double r23542 = x_im;
        double r23543 = y_re;
        double r23544 = r23542 * r23543;
        double r23545 = fma(r23540, r23541, r23544);
        return r23545;
}

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 2019195 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))