Average Error: 0.0 → 0.0
Time: 4.4s
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 r34158 = x_re;
        double r34159 = y_re;
        double r34160 = r34158 * r34159;
        double r34161 = x_im;
        double r34162 = y_im;
        double r34163 = r34161 * r34162;
        double r34164 = r34160 - r34163;
        return r34164;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r34165 = y_im;
        double r34166 = -r34165;
        double r34167 = x_im;
        double r34168 = x_re;
        double r34169 = y_re;
        double r34170 = r34168 * r34169;
        double r34171 = fma(r34166, r34167, r34170);
        return r34171;
}

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 2019196 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))