Average Error: 0.0 → 0.0
Time: 3.0s
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 r1398666 = x_re;
        double r1398667 = y_im;
        double r1398668 = r1398666 * r1398667;
        double r1398669 = x_im;
        double r1398670 = y_re;
        double r1398671 = r1398669 * r1398670;
        double r1398672 = r1398668 + r1398671;
        return r1398672;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1398673 = x_re;
        double r1398674 = y_im;
        double r1398675 = x_im;
        double r1398676 = y_re;
        double r1398677 = r1398675 * r1398676;
        double r1398678 = fma(r1398673, r1398674, r1398677);
        return r1398678;
}

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 2019165 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))