Average Error: 0.0 → 0.0
Time: 7.0s
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 r36263 = x_re;
        double r36264 = y_im;
        double r36265 = r36263 * r36264;
        double r36266 = x_im;
        double r36267 = y_re;
        double r36268 = r36266 * r36267;
        double r36269 = r36265 + r36268;
        return r36269;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r36270 = x_re;
        double r36271 = y_im;
        double r36272 = x_im;
        double r36273 = y_re;
        double r36274 = r36272 * r36273;
        double r36275 = fma(r36270, r36271, r36274);
        return r36275;
}

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 2019195 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))