Average Error: 0.0 → 0.0
Time: 1.0s
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 r91108 = x_re;
        double r91109 = y_im;
        double r91110 = r91108 * r91109;
        double r91111 = x_im;
        double r91112 = y_re;
        double r91113 = r91111 * r91112;
        double r91114 = r91110 + r91113;
        return r91114;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r91115 = y_re;
        double r91116 = x_im;
        double r91117 = y_im;
        double r91118 = x_re;
        double r91119 = r91117 * r91118;
        double r91120 = fma(r91115, r91116, r91119);
        return r91120;
}

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 2020020 +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)))