Average Error: 0.0 → 0.0
Time: 1.9s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1441812 = x_re;
        double r1441813 = y_im;
        double r1441814 = r1441812 * r1441813;
        double r1441815 = x_im;
        double r1441816 = y_re;
        double r1441817 = r1441815 * r1441816;
        double r1441818 = r1441814 + r1441817;
        return r1441818;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1441819 = x_re;
        double r1441820 = y_im;
        double r1441821 = x_im;
        double r1441822 = y_re;
        double r1441823 = r1441821 * r1441822;
        double r1441824 = fma(r1441819, r1441820, r1441823);
        return r1441824;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))