Average Error: 0.0 → 0.0
Time: 1.5s
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 r1276063 = x_re;
        double r1276064 = y_im;
        double r1276065 = r1276063 * r1276064;
        double r1276066 = x_im;
        double r1276067 = y_re;
        double r1276068 = r1276066 * r1276067;
        double r1276069 = r1276065 + r1276068;
        return r1276069;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1276070 = x_re;
        double r1276071 = y_im;
        double r1276072 = x_im;
        double r1276073 = y_re;
        double r1276074 = r1276072 * r1276073;
        double r1276075 = fma(r1276070, r1276071, r1276074);
        return r1276075;
}

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 2019120 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))