Average Error: 0.0 → 0.0
Time: 982.0ms
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r48563 = x_re;
        double r48564 = y_re;
        double r48565 = r48563 * r48564;
        double r48566 = x_im;
        double r48567 = y_im;
        double r48568 = r48566 * r48567;
        double r48569 = r48565 - r48568;
        return r48569;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r48570 = x_re;
        double r48571 = y_re;
        double r48572 = x_im;
        double r48573 = y_im;
        double r48574 = r48572 * r48573;
        double r48575 = -r48574;
        double r48576 = fma(r48570, r48571, r48575);
        return r48576;
}

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

    \[x.re \cdot y.re - x.im \cdot y.im\]
  2. Using strategy rm

  3. Applied fma-neg0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))