Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(-y.im, x.im, x.re \cdot y.re\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(-y.im, x.im, x.re \cdot y.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r37029 = x_re;
        double r37030 = y_re;
        double r37031 = r37029 * r37030;
        double r37032 = x_im;
        double r37033 = y_im;
        double r37034 = r37032 * r37033;
        double r37035 = r37031 - r37034;
        return r37035;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r37036 = y_im;
        double r37037 = -r37036;
        double r37038 = x_im;
        double r37039 = x_re;
        double r37040 = y_re;
        double r37041 = r37039 * r37040;
        double r37042 = fma(r37037, r37038, r37041);
        return r37042;
}

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. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))