Average Error: 0.0 → 0.0
Time: 3.6s
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 r1675770 = x_re;
        double r1675771 = y_im;
        double r1675772 = r1675770 * r1675771;
        double r1675773 = x_im;
        double r1675774 = y_re;
        double r1675775 = r1675773 * r1675774;
        double r1675776 = r1675772 + r1675775;
        return r1675776;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1675777 = x_re;
        double r1675778 = y_im;
        double r1675779 = x_im;
        double r1675780 = y_re;
        double r1675781 = r1675779 * r1675780;
        double r1675782 = fma(r1675777, r1675778, r1675781);
        return r1675782;
}

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 2019163 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))