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 r51149 = x_re;
        double r51150 = y_im;
        double r51151 = r51149 * r51150;
        double r51152 = x_im;
        double r51153 = y_re;
        double r51154 = r51152 * r51153;
        double r51155 = r51151 + r51154;
        return r51155;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r51156 = x_re;
        double r51157 = y_im;
        double r51158 = x_im;
        double r51159 = y_re;
        double r51160 = r51158 * r51159;
        double r51161 = fma(r51156, r51157, r51160);
        return r51161;
}

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 2019362 +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)))