Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1486926 = x_re;
        double r1486927 = y_re;
        double r1486928 = r1486926 * r1486927;
        double r1486929 = x_im;
        double r1486930 = y_im;
        double r1486931 = r1486929 * r1486930;
        double r1486932 = r1486928 - r1486931;
        return r1486932;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1486933 = x_re;
        double r1486934 = y_re;
        double r1486935 = x_im;
        double r1486936 = y_im;
        double r1486937 = r1486935 * r1486936;
        double r1486938 = -r1486937;
        double r1486939 = fma(r1486933, r1486934, r1486938);
        return r1486939;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))