Average Error: 0.0 → 0.0
Time: 5.1s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.im, y.re, \left(x.re \cdot y.im\right)\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.im, y.re, \left(x.re \cdot y.im\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r633202 = x_re;
        double r633203 = y_im;
        double r633204 = r633202 * r633203;
        double r633205 = x_im;
        double r633206 = y_re;
        double r633207 = r633205 * r633206;
        double r633208 = r633204 + r633207;
        return r633208;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r633209 = x_im;
        double r633210 = y_re;
        double r633211 = x_re;
        double r633212 = y_im;
        double r633213 = r633211 * r633212;
        double r633214 = fma(r633209, r633210, r633213);
        return r633214;
}

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

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

  3. Taylor expanded around -inf 0.0

    \[\leadsto \color{blue}{y.re \cdot x.im + y.im \cdot x.re}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))