Average Error: 0.0 → 0.0
Time: 1.7s
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 r2044753 = x_re;
        double r2044754 = y_im;
        double r2044755 = r2044753 * r2044754;
        double r2044756 = x_im;
        double r2044757 = y_re;
        double r2044758 = r2044756 * r2044757;
        double r2044759 = r2044755 + r2044758;
        return r2044759;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2044760 = x_re;
        double r2044761 = y_im;
        double r2044762 = x_im;
        double r2044763 = y_re;
        double r2044764 = r2044762 * r2044763;
        double r2044765 = fma(r2044760, r2044761, r2044764);
        return r2044765;
}

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 2019135 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))