Average Error: 0.0 → 0.0
Time: 7.7s
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 r1099681 = x_re;
        double r1099682 = y_im;
        double r1099683 = r1099681 * r1099682;
        double r1099684 = x_im;
        double r1099685 = y_re;
        double r1099686 = r1099684 * r1099685;
        double r1099687 = r1099683 + r1099686;
        return r1099687;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1099688 = x_re;
        double r1099689 = y_im;
        double r1099690 = x_im;
        double r1099691 = y_re;
        double r1099692 = r1099690 * r1099691;
        double r1099693 = fma(r1099688, r1099689, r1099692);
        return r1099693;
}

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 2019152 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))