Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1489730 = x_re;
        double r1489731 = y_im;
        double r1489732 = r1489730 * r1489731;
        double r1489733 = x_im;
        double r1489734 = y_re;
        double r1489735 = r1489733 * r1489734;
        double r1489736 = r1489732 + r1489735;
        return r1489736;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1489737 = x_re;
        double r1489738 = y_im;
        double r1489739 = x_im;
        double r1489740 = y_re;
        double r1489741 = r1489739 * r1489740;
        double r1489742 = fma(r1489737, r1489738, r1489741);
        return r1489742;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))