Average Error: 0.0 → 0.0
Time: 4.9s
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 r1938440 = x_re;
        double r1938441 = y_im;
        double r1938442 = r1938440 * r1938441;
        double r1938443 = x_im;
        double r1938444 = y_re;
        double r1938445 = r1938443 * r1938444;
        double r1938446 = r1938442 + r1938445;
        return r1938446;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1938447 = x_re;
        double r1938448 = y_im;
        double r1938449 = x_im;
        double r1938450 = y_re;
        double r1938451 = r1938449 * r1938450;
        double r1938452 = fma(r1938447, r1938448, r1938451);
        return r1938452;
}

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 2019149 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))