Average Error: 0.0 → 0.0
Time: 1.0s
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 r49533 = x_re;
        double r49534 = y_im;
        double r49535 = r49533 * r49534;
        double r49536 = x_im;
        double r49537 = y_re;
        double r49538 = r49536 * r49537;
        double r49539 = r49535 + r49538;
        return r49539;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r49540 = y_re;
        double r49541 = x_im;
        double r49542 = y_im;
        double r49543 = x_re;
        double r49544 = r49542 * r49543;
        double r49545 = fma(r49540, r49541, r49544);
        return r49545;
}

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. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020002 +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)))