Average Error: 0.0 → 0.0
Time: 1.1s
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 r113141 = x_re;
        double r113142 = y_im;
        double r113143 = r113141 * r113142;
        double r113144 = x_im;
        double r113145 = y_re;
        double r113146 = r113144 * r113145;
        double r113147 = r113143 + r113146;
        return r113147;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r113148 = x_re;
        double r113149 = y_im;
        double r113150 = x_im;
        double r113151 = y_re;
        double r113152 = r113150 * r113151;
        double r113153 = fma(r113148, r113149, r113152);
        return r113153;
}

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 2019323 +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)))