Average Error: 0.0 → 0.0
Time: 1.1s
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 r34507 = x_re;
        double r34508 = y_im;
        double r34509 = r34507 * r34508;
        double r34510 = x_im;
        double r34511 = y_re;
        double r34512 = r34510 * r34511;
        double r34513 = r34509 + r34512;
        return r34513;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r34514 = x_re;
        double r34515 = y_im;
        double r34516 = x_im;
        double r34517 = y_re;
        double r34518 = r34516 * r34517;
        double r34519 = fma(r34514, r34515, r34518);
        return r34519;
}

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 2020001 +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)))