Average Error: 0.0 → 0.0
Time: 2.9s
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 r109899 = x_re;
        double r109900 = y_re;
        double r109901 = r109899 * r109900;
        double r109902 = x_im;
        double r109903 = y_im;
        double r109904 = r109902 * r109903;
        double r109905 = r109901 - r109904;
        return r109905;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r109906 = x_re;
        double r109907 = y_re;
        double r109908 = x_im;
        double r109909 = y_im;
        double r109910 = r109908 * r109909;
        double r109911 = -r109910;
        double r109912 = fma(r109906, r109907, r109911);
        return r109912;
}

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 2020003 +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)))