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 r24266 = x_re;
        double r24267 = y_im;
        double r24268 = r24266 * r24267;
        double r24269 = x_im;
        double r24270 = y_re;
        double r24271 = r24269 * r24270;
        double r24272 = r24268 + r24271;
        return r24272;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r24273 = x_re;
        double r24274 = y_im;
        double r24275 = x_im;
        double r24276 = y_re;
        double r24277 = r24275 * r24276;
        double r24278 = fma(r24273, r24274, r24277);
        return r24278;
}

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 2020018 +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)))