Average Error: 0.0 → 0.0
Time: 4.9s
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 r1829298 = x_re;
        double r1829299 = y_im;
        double r1829300 = r1829298 * r1829299;
        double r1829301 = x_im;
        double r1829302 = y_re;
        double r1829303 = r1829301 * r1829302;
        double r1829304 = r1829300 + r1829303;
        return r1829304;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1829305 = x_im;
        double r1829306 = y_re;
        double r1829307 = x_re;
        double r1829308 = y_im;
        double r1829309 = r1829307 * r1829308;
        double r1829310 = fma(r1829305, r1829306, r1829309);
        return r1829310;
}

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 2019132 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))