Average Error: 0.0 → 0.0
Time: 1.9s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(y.re, x.im, \left(y.im \cdot x.re\right)\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(y.re, x.im, \left(y.im \cdot x.re\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1137019 = x_re;
        double r1137020 = y_im;
        double r1137021 = r1137019 * r1137020;
        double r1137022 = x_im;
        double r1137023 = y_re;
        double r1137024 = r1137022 * r1137023;
        double r1137025 = r1137021 + r1137024;
        return r1137025;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1137026 = y_re;
        double r1137027 = x_im;
        double r1137028 = y_im;
        double r1137029 = x_re;
        double r1137030 = r1137028 * r1137029;
        double r1137031 = fma(r1137026, r1137027, r1137030);
        return r1137031;
}

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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))