Average Error: 0.0 → 0.0
Time: 1.5s
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 r1294582 = x_re;
        double r1294583 = y_im;
        double r1294584 = r1294582 * r1294583;
        double r1294585 = x_im;
        double r1294586 = y_re;
        double r1294587 = r1294585 * r1294586;
        double r1294588 = r1294584 + r1294587;
        return r1294588;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1294589 = x_re;
        double r1294590 = y_im;
        double r1294591 = x_im;
        double r1294592 = y_re;
        double r1294593 = r1294591 * r1294592;
        double r1294594 = fma(r1294589, r1294590, r1294593);
        return r1294594;
}

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 2019158 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))