Average Error: 0.0 → 0.0
Time: 980.0ms
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 r44179 = x_re;
        double r44180 = y_im;
        double r44181 = r44179 * r44180;
        double r44182 = x_im;
        double r44183 = y_re;
        double r44184 = r44182 * r44183;
        double r44185 = r44181 + r44184;
        return r44185;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r44186 = x_re;
        double r44187 = y_im;
        double r44188 = x_im;
        double r44189 = y_re;
        double r44190 = r44188 * r44189;
        double r44191 = fma(r44186, r44187, r44190);
        return r44191;
}

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 2020036 +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)))