Average Error: 0.0 → 0.0
Time: 746.0ms
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 r44211 = x_re;
        double r44212 = y_im;
        double r44213 = r44211 * r44212;
        double r44214 = x_im;
        double r44215 = y_re;
        double r44216 = r44214 * r44215;
        double r44217 = r44213 + r44216;
        return r44217;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r44218 = x_re;
        double r44219 = y_im;
        double r44220 = x_im;
        double r44221 = y_re;
        double r44222 = r44220 * r44221;
        double r44223 = fma(r44218, r44219, r44222);
        return r44223;
}

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 2020033 +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)))