Average Error: 0.0 → 0.0
Time: 1.1s
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 r89524 = x_re;
        double r89525 = y_re;
        double r89526 = r89524 * r89525;
        double r89527 = x_im;
        double r89528 = y_im;
        double r89529 = r89527 * r89528;
        double r89530 = r89526 - r89529;
        return r89530;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r89531 = x_re;
        double r89532 = y_re;
        double r89533 = x_im;
        double r89534 = y_im;
        double r89535 = r89533 * r89534;
        double r89536 = -r89535;
        double r89537 = fma(r89531, r89532, r89536);
        return r89537;
}

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 2019353 +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)))