Average Error: 0.0 → 0.0
Time: 5.9s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.im, y.re, x.re \cdot y.im\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.im, y.re, x.re \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2567548 = x_re;
        double r2567549 = y_im;
        double r2567550 = r2567548 * r2567549;
        double r2567551 = x_im;
        double r2567552 = y_re;
        double r2567553 = r2567551 * r2567552;
        double r2567554 = r2567550 + r2567553;
        return r2567554;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2567555 = x_im;
        double r2567556 = y_re;
        double r2567557 = x_re;
        double r2567558 = y_im;
        double r2567559 = r2567557 * r2567558;
        double r2567560 = fma(r2567555, r2567556, r2567559);
        return r2567560;
}

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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))