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 r777997 = x_re;
        double r777998 = y_re;
        double r777999 = r777997 * r777998;
        double r778000 = x_im;
        double r778001 = y_im;
        double r778002 = r778000 * r778001;
        double r778003 = r777999 - r778002;
        return r778003;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r778004 = x_re;
        double r778005 = y_re;
        double r778006 = x_im;
        double r778007 = y_im;
        double r778008 = r778006 * r778007;
        double r778009 = -r778008;
        double r778010 = fma(r778004, r778005, r778009);
        return r778010;
}

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 2019120 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))