Average Error: 0.0 → 0.0
Time: 3.6s
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 r50145 = x_re;
        double r50146 = y_re;
        double r50147 = r50145 * r50146;
        double r50148 = x_im;
        double r50149 = y_im;
        double r50150 = r50148 * r50149;
        double r50151 = r50147 - r50150;
        return r50151;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r50152 = y_im;
        double r50153 = -r50152;
        double r50154 = x_im;
        double r50155 = x_re;
        double r50156 = y_re;
        double r50157 = r50155 * r50156;
        double r50158 = fma(r50153, r50154, r50157);
        return r50158;
}

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 2019174 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))