Average Error: 0.0 → 0.0
Time: 17.0s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.im, y.re, y.im \cdot x.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.im, y.re, y.im \cdot x.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2243642 = x_re;
        double r2243643 = y_im;
        double r2243644 = r2243642 * r2243643;
        double r2243645 = x_im;
        double r2243646 = y_re;
        double r2243647 = r2243645 * r2243646;
        double r2243648 = r2243644 + r2243647;
        return r2243648;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2243649 = x_im;
        double r2243650 = y_re;
        double r2243651 = y_im;
        double r2243652 = x_re;
        double r2243653 = r2243651 * r2243652;
        double r2243654 = fma(r2243649, r2243650, r2243653);
        return r2243654;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))