Average Error: 0.0 → 0.0
Time: 5.0s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r42426 = x_re;
        double r42427 = y_im;
        double r42428 = r42426 * r42427;
        double r42429 = x_im;
        double r42430 = y_re;
        double r42431 = r42429 * r42430;
        double r42432 = r42428 + r42431;
        return r42432;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r42433 = y_re;
        double r42434 = x_im;
        double r42435 = y_im;
        double r42436 = x_re;
        double r42437 = r42435 * r42436;
        double r42438 = fma(r42433, r42434, r42437);
        return r42438;
}

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. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{y.re \cdot x.im + y.im \cdot x.re}\]

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019306 +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)))