Average Error: 0.0 → 0.0
Time: 1.2s
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 r39822 = x_re;
        double r39823 = y_im;
        double r39824 = r39822 * r39823;
        double r39825 = x_im;
        double r39826 = y_re;
        double r39827 = r39825 * r39826;
        double r39828 = r39824 + r39827;
        return r39828;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r39829 = x_re;
        double r39830 = y_im;
        double r39831 = x_im;
        double r39832 = y_re;
        double r39833 = r39831 * r39832;
        double r39834 = fma(r39829, r39830, r39833);
        return r39834;
}

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 2020056 +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)))