Average Error: 0.0 → 0.0
Time: 7.3s
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 r3073622 = x_re;
        double r3073623 = y_im;
        double r3073624 = r3073622 * r3073623;
        double r3073625 = x_im;
        double r3073626 = y_re;
        double r3073627 = r3073625 * r3073626;
        double r3073628 = r3073624 + r3073627;
        return r3073628;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r3073629 = x_im;
        double r3073630 = y_re;
        double r3073631 = x_re;
        double r3073632 = y_im;
        double r3073633 = r3073631 * r3073632;
        double r3073634 = fma(r3073629, r3073630, r3073633);
        return r3073634;
}

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 2019169 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))