Average Error: 0.0 → 0.0
Time: 5.6s
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 r1296666 = x_re;
        double r1296667 = y_im;
        double r1296668 = r1296666 * r1296667;
        double r1296669 = x_im;
        double r1296670 = y_re;
        double r1296671 = r1296669 * r1296670;
        double r1296672 = r1296668 + r1296671;
        return r1296672;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1296673 = x_re;
        double r1296674 = y_im;
        double r1296675 = x_im;
        double r1296676 = y_re;
        double r1296677 = r1296675 * r1296676;
        double r1296678 = fma(r1296673, r1296674, r1296677);
        return r1296678;
}

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 2019144 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))