Average Error: 0.0 → 0.0
Time: 1.0s
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 r43201 = x_re;
        double r43202 = y_im;
        double r43203 = r43201 * r43202;
        double r43204 = x_im;
        double r43205 = y_re;
        double r43206 = r43204 * r43205;
        double r43207 = r43203 + r43206;
        return r43207;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r43208 = x_re;
        double r43209 = y_im;
        double r43210 = x_im;
        double r43211 = y_re;
        double r43212 = r43210 * r43211;
        double r43213 = fma(r43208, r43209, r43212);
        return r43213;
}

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 2020056 +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)))