Average Error: 0.0 → 0.0
Time: 1.5s
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 r3005510 = x_re;
        double r3005511 = y_im;
        double r3005512 = r3005510 * r3005511;
        double r3005513 = x_im;
        double r3005514 = y_re;
        double r3005515 = r3005513 * r3005514;
        double r3005516 = r3005512 + r3005515;
        return r3005516;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r3005517 = x_re;
        double r3005518 = y_im;
        double r3005519 = x_im;
        double r3005520 = y_re;
        double r3005521 = r3005519 * r3005520;
        double r3005522 = fma(r3005517, r3005518, r3005521);
        return r3005522;
}

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 2019179 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))