Average Error: 0.0 → 0.0
Time: 1.5s
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 r1707288 = x_re;
        double r1707289 = y_im;
        double r1707290 = r1707288 * r1707289;
        double r1707291 = x_im;
        double r1707292 = y_re;
        double r1707293 = r1707291 * r1707292;
        double r1707294 = r1707290 + r1707293;
        return r1707294;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1707295 = x_re;
        double r1707296 = y_im;
        double r1707297 = x_im;
        double r1707298 = y_re;
        double r1707299 = r1707297 * r1707298;
        double r1707300 = fma(r1707295, r1707296, r1707299);
        return r1707300;
}

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 2019165 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))