Average Error: 0.0 → 0.0
Time: 1.6s
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 r862153 = x_re;
        double r862154 = y_im;
        double r862155 = r862153 * r862154;
        double r862156 = x_im;
        double r862157 = y_re;
        double r862158 = r862156 * r862157;
        double r862159 = r862155 + r862158;
        return r862159;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r862160 = x_re;
        double r862161 = y_im;
        double r862162 = x_im;
        double r862163 = y_re;
        double r862164 = r862162 * r862163;
        double r862165 = fma(r862160, r862161, r862164);
        return r862165;
}

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 2019139 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))