Average Error: 0.0 → 0.0
Time: 2.8s
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 r1665177 = x_re;
        double r1665178 = y_im;
        double r1665179 = r1665177 * r1665178;
        double r1665180 = x_im;
        double r1665181 = y_re;
        double r1665182 = r1665180 * r1665181;
        double r1665183 = r1665179 + r1665182;
        return r1665183;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1665184 = y_re;
        double r1665185 = x_im;
        double r1665186 = y_im;
        double r1665187 = x_re;
        double r1665188 = r1665186 * r1665187;
        double r1665189 = fma(r1665184, r1665185, r1665188);
        return r1665189;
}

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 2019128 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))