Average Error: 0.0 → 0.0
Time: 4.3s
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 r64773 = x_re;
        double r64774 = y_re;
        double r64775 = r64773 * r64774;
        double r64776 = x_im;
        double r64777 = y_im;
        double r64778 = r64776 * r64777;
        double r64779 = r64775 - r64778;
        return r64779;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r64780 = x_re;
        double r64781 = y_re;
        double r64782 = x_im;
        double r64783 = y_im;
        double r64784 = r64782 * r64783;
        double r64785 = -r64784;
        double r64786 = fma(r64780, r64781, r64785);
        return r64786;
}

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 2019346 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))