Average Error: 0.0 → 0.0
Time: 11.6s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1234622 = x_re;
        double r1234623 = y_re;
        double r1234624 = r1234622 * r1234623;
        double r1234625 = x_im;
        double r1234626 = y_im;
        double r1234627 = r1234625 * r1234626;
        double r1234628 = r1234624 - r1234627;
        return r1234628;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1234629 = x_re;
        double r1234630 = y_re;
        double r1234631 = x_im;
        double r1234632 = y_im;
        double r1234633 = r1234631 * r1234632;
        double r1234634 = -r1234633;
        double r1234635 = fma(r1234629, r1234630, r1234634);
        return r1234635;
}

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

    \[x.re \cdot y.re - x.im \cdot y.im\]
  2. Using strategy rm

  3. Applied fma-neg0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))