Average Error: 0.0 → 0.0
Time: 7.0s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(y.re, x.im, x.re \cdot y.im\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(y.re, x.im, x.re \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r4291167 = x_re;
        double r4291168 = y_im;
        double r4291169 = r4291167 * r4291168;
        double r4291170 = x_im;
        double r4291171 = y_re;
        double r4291172 = r4291170 * r4291171;
        double r4291173 = r4291169 + r4291172;
        return r4291173;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r4291174 = y_re;
        double r4291175 = x_im;
        double r4291176 = x_re;
        double r4291177 = y_im;
        double r4291178 = r4291176 * r4291177;
        double r4291179 = fma(r4291174, r4291175, r4291178);
        return r4291179;
}

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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))