Average Error: 0.0 → 0.0
Time: 1.0s
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 r38766 = x_re;
        double r38767 = y_re;
        double r38768 = r38766 * r38767;
        double r38769 = x_im;
        double r38770 = y_im;
        double r38771 = r38769 * r38770;
        double r38772 = r38768 - r38771;
        return r38772;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r38773 = x_re;
        double r38774 = y_re;
        double r38775 = x_im;
        double r38776 = y_im;
        double r38777 = r38775 * r38776;
        double r38778 = -r38777;
        double r38779 = fma(r38773, r38774, r38778);
        return r38779;
}

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 2020064 +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)))