Average Error: 0.0 → 0.0
Time: 841.0ms
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 r47172 = x_re;
        double r47173 = y_re;
        double r47174 = r47172 * r47173;
        double r47175 = x_im;
        double r47176 = y_im;
        double r47177 = r47175 * r47176;
        double r47178 = r47174 - r47177;
        return r47178;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r47179 = x_re;
        double r47180 = y_re;
        double r47181 = x_im;
        double r47182 = y_im;
        double r47183 = r47181 * r47182;
        double r47184 = -r47183;
        double r47185 = fma(r47179, r47180, r47184);
        return r47185;
}

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 2020021 +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)))