Average Error: 0.0 → 0.0
Time: 1.2s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r40009 = x_re;
        double r40010 = y_im;
        double r40011 = r40009 * r40010;
        double r40012 = x_im;
        double r40013 = y_re;
        double r40014 = r40012 * r40013;
        double r40015 = r40011 + r40014;
        return r40015;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r40016 = y_re;
        double r40017 = x_im;
        double r40018 = y_im;
        double r40019 = x_re;
        double r40020 = r40018 * r40019;
        double r40021 = fma(r40016, r40017, r40020);
        return r40021;
}

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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020089 +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)))