Average Error: 0.0 → 0.0
Time: 3.1s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.im, y.re, \left(x.re \cdot y.im\right)\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.im, y.re, \left(x.re \cdot y.im\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r900984 = x_re;
        double r900985 = y_im;
        double r900986 = r900984 * r900985;
        double r900987 = x_im;
        double r900988 = y_re;
        double r900989 = r900987 * r900988;
        double r900990 = r900986 + r900989;
        return r900990;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r900991 = x_im;
        double r900992 = y_re;
        double r900993 = x_re;
        double r900994 = y_im;
        double r900995 = r900993 * r900994;
        double r900996 = fma(r900991, r900992, r900995);
        return r900996;
}

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. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{y.re \cdot x.im + y.im \cdot x.re}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))