Average Error: 0.0 → 0.0
Time: 7.9s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, -y.im \cdot x.im\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, -y.im \cdot x.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r33001 = x_re;
        double r33002 = y_re;
        double r33003 = r33001 * r33002;
        double r33004 = x_im;
        double r33005 = y_im;
        double r33006 = r33004 * r33005;
        double r33007 = r33003 - r33006;
        return r33007;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r33008 = x_re;
        double r33009 = y_re;
        double r33010 = y_im;
        double r33011 = x_im;
        double r33012 = r33010 * r33011;
        double r33013 = -r33012;
        double r33014 = fma(r33008, r33009, r33013);
        return r33014;
}

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. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))