Average Error: 0.0 → 0.0
Time: 2.6s
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 r44499 = x_re;
        double r44500 = y_re;
        double r44501 = r44499 * r44500;
        double r44502 = x_im;
        double r44503 = y_im;
        double r44504 = r44502 * r44503;
        double r44505 = r44501 - r44504;
        return r44505;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r44506 = x_re;
        double r44507 = y_re;
        double r44508 = x_im;
        double r44509 = y_im;
        double r44510 = r44508 * r44509;
        double r44511 = -r44510;
        double r44512 = fma(r44506, r44507, r44511);
        return r44512;
}

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 2020024 +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)))