Average Error: 0.0 → 0.0
Time: 891.0ms
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 r54715 = x_re;
        double r54716 = y_re;
        double r54717 = r54715 * r54716;
        double r54718 = x_im;
        double r54719 = y_im;
        double r54720 = r54718 * r54719;
        double r54721 = r54717 - r54720;
        return r54721;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r54722 = x_re;
        double r54723 = y_re;
        double r54724 = x_im;
        double r54725 = y_im;
        double r54726 = r54724 * r54725;
        double r54727 = -r54726;
        double r54728 = fma(r54722, r54723, r54727);
        return r54728;
}

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 2020001 +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)))