Average Error: 0.0 → 0.0
Time: 4.6s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, \left(x.im \cdot y.re\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1096039 = x_re;
        double r1096040 = y_im;
        double r1096041 = r1096039 * r1096040;
        double r1096042 = x_im;
        double r1096043 = y_re;
        double r1096044 = r1096042 * r1096043;
        double r1096045 = r1096041 + r1096044;
        return r1096045;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1096046 = x_re;
        double r1096047 = y_im;
        double r1096048 = x_im;
        double r1096049 = y_re;
        double r1096050 = r1096048 * r1096049;
        double r1096051 = fma(r1096046, r1096047, r1096050);
        return r1096051;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))