_multiplyComplex, imaginary part

Average Error: 0.0 → 0.0

Time: 4.1s

Precision: 64

Math

\[x.re \cdot y.im + x.im \cdot y.re\]

↓

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

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r78739 = x_re;
        double r78740 = y_im;
        double r78741 = r78739 * r78740;
        double r78742 = x_im;
        double r78743 = y_re;
        double r78744 = r78742 * r78743;
        double r78745 = r78741 + r78744;
        return r78745;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r78746 = x_re;
        double r78747 = y_im;
        double r78748 = x_im;
        double r78749 = y_re;
        double r78750 = r78748 * r78749;
        double r78751 = fma(r78746, r78747, r78750);
        return r78751;
}

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. Simplified 0.0

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

3. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019325 +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)))