Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2439710 = x_re;
        double r2439711 = y_re;
        double r2439712 = r2439710 * r2439711;
        double r2439713 = x_im;
        double r2439714 = y_im;
        double r2439715 = r2439713 * r2439714;
        double r2439716 = r2439712 - r2439715;
        return r2439716;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2439717 = x_re;
        double r2439718 = y_re;
        double r2439719 = x_im;
        double r2439720 = y_im;
        double r2439721 = r2439719 * r2439720;
        double r2439722 = -r2439721;
        double r2439723 = fma(r2439717, r2439718, r2439722);
        return r2439723;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))