Average Error: 0.0 → 0.0
Time: 3.0s
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 r53355 = x_re;
        double r53356 = y_re;
        double r53357 = r53355 * r53356;
        double r53358 = x_im;
        double r53359 = y_im;
        double r53360 = r53358 * r53359;
        double r53361 = r53357 - r53360;
        return r53361;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r53362 = x_re;
        double r53363 = y_re;
        double r53364 = x_im;
        double r53365 = y_im;
        double r53366 = r53364 * r53365;
        double r53367 = -r53366;
        double r53368 = fma(r53362, r53363, r53367);
        return r53368;
}

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 2019354 +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)))