Average Error: 0.0 → 0.0
Time: 675.0ms
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 r44821 = x_re;
        double r44822 = y_im;
        double r44823 = r44821 * r44822;
        double r44824 = x_im;
        double r44825 = y_re;
        double r44826 = r44824 * r44825;
        double r44827 = r44823 + r44826;
        return r44827;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r44828 = x_re;
        double r44829 = y_im;
        double r44830 = x_im;
        double r44831 = y_re;
        double r44832 = r44830 * r44831;
        double r44833 = fma(r44828, r44829, r44832);
        return r44833;
}

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 2020062 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))