Average Error: 0.0 → 0.0
Time: 1.5s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1752641 = x_re;
        double r1752642 = y_im;
        double r1752643 = r1752641 * r1752642;
        double r1752644 = x_im;
        double r1752645 = y_re;
        double r1752646 = r1752644 * r1752645;
        double r1752647 = r1752643 + r1752646;
        return r1752647;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1752648 = x_re;
        double r1752649 = y_im;
        double r1752650 = x_im;
        double r1752651 = y_re;
        double r1752652 = r1752650 * r1752651;
        double r1752653 = fma(r1752648, r1752649, r1752652);
        return r1752653;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))