Average Error: 0.0 → 0.0
Time: 8.2s
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 r1829537 = x_re;
        double r1829538 = y_im;
        double r1829539 = r1829537 * r1829538;
        double r1829540 = x_im;
        double r1829541 = y_re;
        double r1829542 = r1829540 * r1829541;
        double r1829543 = r1829539 + r1829542;
        return r1829543;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1829544 = x_re;
        double r1829545 = y_im;
        double r1829546 = x_im;
        double r1829547 = y_re;
        double r1829548 = r1829546 * r1829547;
        double r1829549 = fma(r1829544, r1829545, r1829548);
        return r1829549;
}

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 2019162 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))