Average Error: 0.0 → 0.0
Time: 1.8s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2668521 = x_re;
        double r2668522 = y_im;
        double r2668523 = r2668521 * r2668522;
        double r2668524 = x_im;
        double r2668525 = y_re;
        double r2668526 = r2668524 * r2668525;
        double r2668527 = r2668523 + r2668526;
        return r2668527;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2668528 = x_re;
        double r2668529 = y_im;
        double r2668530 = x_im;
        double r2668531 = y_re;
        double r2668532 = r2668530 * r2668531;
        double r2668533 = fma(r2668528, r2668529, r2668532);
        return r2668533;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))