Average Error: 0.0 → 0.0
Time: 5.4s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1548477 = x_re;
        double r1548478 = y_im;
        double r1548479 = r1548477 * r1548478;
        double r1548480 = x_im;
        double r1548481 = y_re;
        double r1548482 = r1548480 * r1548481;
        double r1548483 = r1548479 + r1548482;
        return r1548483;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1548484 = x_re;
        double r1548485 = y_im;
        double r1548486 = x_im;
        double r1548487 = y_re;
        double r1548488 = r1548486 * r1548487;
        double r1548489 = fma(r1548484, r1548485, r1548488);
        return r1548489;
}

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

    \[x.re \cdot y.im + x.im \cdot y.re\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))