Average Error: 0.0 → 0.0
Time: 1.2s
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 r1426007 = x_re;
        double r1426008 = y_im;
        double r1426009 = r1426007 * r1426008;
        double r1426010 = x_im;
        double r1426011 = y_re;
        double r1426012 = r1426010 * r1426011;
        double r1426013 = r1426009 + r1426012;
        return r1426013;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1426014 = x_re;
        double r1426015 = y_im;
        double r1426016 = x_im;
        double r1426017 = y_re;
        double r1426018 = r1426016 * r1426017;
        double r1426019 = fma(r1426014, r1426015, r1426018);
        return r1426019;
}

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 2019168 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))