Average Error: 0.0 → 0.0
Time: 1.3s
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 r44546 = x_re;
        double r44547 = y_im;
        double r44548 = r44546 * r44547;
        double r44549 = x_im;
        double r44550 = y_re;
        double r44551 = r44549 * r44550;
        double r44552 = r44548 + r44551;
        return r44552;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r44553 = x_re;
        double r44554 = y_im;
        double r44555 = x_im;
        double r44556 = y_re;
        double r44557 = r44555 * r44556;
        double r44558 = fma(r44553, r44554, r44557);
        return r44558;
}

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 2020034 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))