Average Error: 0.0 → 0.0
Time: 3.5s
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 r1433395 = x_re;
        double r1433396 = y_im;
        double r1433397 = r1433395 * r1433396;
        double r1433398 = x_im;
        double r1433399 = y_re;
        double r1433400 = r1433398 * r1433399;
        double r1433401 = r1433397 + r1433400;
        return r1433401;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1433402 = x_im;
        double r1433403 = y_re;
        double r1433404 = x_re;
        double r1433405 = y_im;
        double r1433406 = r1433404 * r1433405;
        double r1433407 = fma(r1433402, r1433403, r1433406);
        return r1433407;
}

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 2019151 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))