Average Error: 0.0 → 0.0
Time: 7.0s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.im, y.re, x.re \cdot y.im\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.im, y.re, x.re \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1540746 = x_re;
        double r1540747 = y_im;
        double r1540748 = r1540746 * r1540747;
        double r1540749 = x_im;
        double r1540750 = y_re;
        double r1540751 = r1540749 * r1540750;
        double r1540752 = r1540748 + r1540751;
        return r1540752;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1540753 = x_im;
        double r1540754 = y_re;
        double r1540755 = x_re;
        double r1540756 = y_im;
        double r1540757 = r1540755 * r1540756;
        double r1540758 = fma(r1540753, r1540754, r1540757);
        return r1540758;
}

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. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))