Average Error: 0.0 → 0.0
Time: 5.4s
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 r1277775 = x_re;
        double r1277776 = y_im;
        double r1277777 = r1277775 * r1277776;
        double r1277778 = x_im;
        double r1277779 = y_re;
        double r1277780 = r1277778 * r1277779;
        double r1277781 = r1277777 + r1277780;
        return r1277781;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1277782 = x_re;
        double r1277783 = y_im;
        double r1277784 = x_im;
        double r1277785 = y_re;
        double r1277786 = r1277784 * r1277785;
        double r1277787 = fma(r1277782, r1277783, r1277786);
        return r1277787;
}

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 2019129 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))