Average Error: 0.0 → 0.0
Time: 7.3s
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 r1931041 = x_re;
        double r1931042 = y_im;
        double r1931043 = r1931041 * r1931042;
        double r1931044 = x_im;
        double r1931045 = y_re;
        double r1931046 = r1931044 * r1931045;
        double r1931047 = r1931043 + r1931046;
        return r1931047;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1931048 = x_re;
        double r1931049 = y_im;
        double r1931050 = x_im;
        double r1931051 = y_re;
        double r1931052 = r1931050 * r1931051;
        double r1931053 = fma(r1931048, r1931049, r1931052);
        return r1931053;
}

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 2019162 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))