Average Error: 0.0 → 0.0
Time: 3.8s
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 r105846 = x_re;
        double r105847 = y_im;
        double r105848 = r105846 * r105847;
        double r105849 = x_im;
        double r105850 = y_re;
        double r105851 = r105849 * r105850;
        double r105852 = r105848 + r105851;
        return r105852;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r105853 = x_re;
        double r105854 = y_im;
        double r105855 = x_im;
        double r105856 = y_re;
        double r105857 = r105855 * r105856;
        double r105858 = fma(r105853, r105854, r105857);
        return r105858;
}

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 2019325 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))