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 r88928 = x_re;
        double r88929 = y_im;
        double r88930 = r88928 * r88929;
        double r88931 = x_im;
        double r88932 = y_re;
        double r88933 = r88931 * r88932;
        double r88934 = r88930 + r88933;
        return r88934;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r88935 = x_re;
        double r88936 = y_im;
        double r88937 = x_im;
        double r88938 = y_re;
        double r88939 = r88937 * r88938;
        double r88940 = fma(r88935, r88936, r88939);
        return r88940;
}

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 2020001 +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)))