Average Error: 0.0 → 0.0
Time: 11.3s
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 r32735 = x_re;
        double r32736 = y_re;
        double r32737 = r32735 * r32736;
        double r32738 = x_im;
        double r32739 = y_im;
        double r32740 = r32738 * r32739;
        double r32741 = r32737 - r32740;
        return r32741;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r32742 = x_re;
        double r32743 = y_re;
        double r32744 = y_im;
        double r32745 = x_im;
        double r32746 = r32744 * r32745;
        double r32747 = -r32746;
        double r32748 = fma(r32742, r32743, r32747);
        return r32748;
}

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 2019306 +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)))