Average Error: 0.0 → 0.0
Time: 5.3s
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 r1238482 = x_re;
        double r1238483 = y_im;
        double r1238484 = r1238482 * r1238483;
        double r1238485 = x_im;
        double r1238486 = y_re;
        double r1238487 = r1238485 * r1238486;
        double r1238488 = r1238484 + r1238487;
        return r1238488;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1238489 = x_re;
        double r1238490 = y_im;
        double r1238491 = x_im;
        double r1238492 = y_re;
        double r1238493 = r1238491 * r1238492;
        double r1238494 = fma(r1238489, r1238490, r1238493);
        return r1238494;
}

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 2019146 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))