Average Error: 0.0 → 0.0
Time: 4.0s
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 r2304499 = x_re;
        double r2304500 = y_im;
        double r2304501 = r2304499 * r2304500;
        double r2304502 = x_im;
        double r2304503 = y_re;
        double r2304504 = r2304502 * r2304503;
        double r2304505 = r2304501 + r2304504;
        return r2304505;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2304506 = x_re;
        double r2304507 = y_im;
        double r2304508 = x_im;
        double r2304509 = y_re;
        double r2304510 = r2304508 * r2304509;
        double r2304511 = fma(r2304506, r2304507, r2304510);
        return r2304511;
}

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 2019163 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))