Average Error: 0.0 → 0.0
Time: 6.9s
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 r40477 = x_re;
        double r40478 = y_re;
        double r40479 = r40477 * r40478;
        double r40480 = x_im;
        double r40481 = y_im;
        double r40482 = r40480 * r40481;
        double r40483 = r40479 - r40482;
        return r40483;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r40484 = y_im;
        double r40485 = -r40484;
        double r40486 = x_im;
        double r40487 = x_re;
        double r40488 = y_re;
        double r40489 = r40487 * r40488;
        double r40490 = fma(r40485, r40486, r40489);
        return r40490;
}

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 2019194 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))