Average Error: 0.0 → 0.0
Time: 709.0ms
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 r108089 = x_re;
        double r108090 = y_im;
        double r108091 = r108089 * r108090;
        double r108092 = x_im;
        double r108093 = y_re;
        double r108094 = r108092 * r108093;
        double r108095 = r108091 + r108094;
        return r108095;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r108096 = x_re;
        double r108097 = y_im;
        double r108098 = x_im;
        double r108099 = y_re;
        double r108100 = r108098 * r108099;
        double r108101 = fma(r108096, r108097, r108100);
        return r108101;
}

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 2019353 +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)))