Average Error: 0.0 → 0.0
Time: 1.4s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r50877 = x_re;
        double r50878 = y_im;
        double r50879 = r50877 * r50878;
        double r50880 = x_im;
        double r50881 = y_re;
        double r50882 = r50880 * r50881;
        double r50883 = r50879 + r50882;
        return r50883;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r50884 = y_re;
        double r50885 = x_im;
        double r50886 = y_im;
        double r50887 = x_re;
        double r50888 = r50886 * r50887;
        double r50889 = fma(r50884, r50885, r50888);
        return r50889;
}

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. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{y.re \cdot x.im + y.im \cdot x.re}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)\]

Reproduce

herbie shell --seed 2019352 +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)))