Average Error: 0.0 → 0.0
Time: 5.8s
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 r30984 = x_re;
        double r30985 = y_im;
        double r30986 = r30984 * r30985;
        double r30987 = x_im;
        double r30988 = y_re;
        double r30989 = r30987 * r30988;
        double r30990 = r30986 + r30989;
        return r30990;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r30991 = x_re;
        double r30992 = y_im;
        double r30993 = x_im;
        double r30994 = y_re;
        double r30995 = r30993 * r30994;
        double r30996 = fma(r30991, r30992, r30995);
        return r30996;
}

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 2019235 +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)))