Average Error: 0.0 → 0.0
Time: 5.5s
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 r1496103 = x_re;
        double r1496104 = y_im;
        double r1496105 = r1496103 * r1496104;
        double r1496106 = x_im;
        double r1496107 = y_re;
        double r1496108 = r1496106 * r1496107;
        double r1496109 = r1496105 + r1496108;
        return r1496109;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1496110 = x_re;
        double r1496111 = y_im;
        double r1496112 = x_im;
        double r1496113 = y_re;
        double r1496114 = r1496112 * r1496113;
        double r1496115 = fma(r1496110, r1496111, r1496114);
        return r1496115;
}

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 2019142 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))