Average Error: 0.0 → 0.0
Time: 835.0ms
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 r39980 = x_re;
        double r39981 = y_im;
        double r39982 = r39980 * r39981;
        double r39983 = x_im;
        double r39984 = y_re;
        double r39985 = r39983 * r39984;
        double r39986 = r39982 + r39985;
        return r39986;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r39987 = y_re;
        double r39988 = x_im;
        double r39989 = y_im;
        double r39990 = x_re;
        double r39991 = r39989 * r39990;
        double r39992 = fma(r39987, r39988, r39991);
        return r39992;
}

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. Taylor expanded around inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020027 +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)))