Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(y.re, x.im, x.re \cdot y.im\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(y.re, x.im, x.re \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r55988 = x_re;
        double r55989 = y_im;
        double r55990 = r55988 * r55989;
        double r55991 = x_im;
        double r55992 = y_re;
        double r55993 = r55991 * r55992;
        double r55994 = r55990 + r55993;
        return r55994;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r55995 = y_re;
        double r55996 = x_im;
        double r55997 = x_re;
        double r55998 = y_im;
        double r55999 = r55997 * r55998;
        double r56000 = fma(r55995, r55996, r55999);
        return r56000;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020042 +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)))