Average Error: 0.0 → 0.0
Time: 3.3s
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 r248 = x_re;
        double r249 = y_re;
        double r250 = r248 * r249;
        double r251 = x_im;
        double r252 = y_im;
        double r253 = r251 * r252;
        double r254 = r250 - r253;
        return r254;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r255 = x_re;
        double r256 = y_re;
        double r257 = x_im;
        double r258 = y_im;
        double r259 = r257 * r258;
        double r260 = -r259;
        double r261 = fma(r255, r256, r260);
        return r261;
}

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 2020025 +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)))