Average Error: 0.0 → 0.0
Time: 7.0s
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 r44674 = x_re;
        double r44675 = y_re;
        double r44676 = r44674 * r44675;
        double r44677 = x_im;
        double r44678 = y_im;
        double r44679 = r44677 * r44678;
        double r44680 = r44676 - r44679;
        return r44680;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r44681 = x_re;
        double r44682 = y_re;
        double r44683 = y_im;
        double r44684 = x_im;
        double r44685 = r44683 * r44684;
        double r44686 = -r44685;
        double r44687 = fma(r44681, r44682, r44686);
        return r44687;
}

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 2019326 +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)))