Average Error: 0.0 → 0.0
Time: 8.2s
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 r109588 = x_re;
        double r109589 = y_re;
        double r109590 = r109588 * r109589;
        double r109591 = x_im;
        double r109592 = y_im;
        double r109593 = r109591 * r109592;
        double r109594 = r109590 - r109593;
        return r109594;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r109595 = x_re;
        double r109596 = y_re;
        double r109597 = y_im;
        double r109598 = x_im;
        double r109599 = r109597 * r109598;
        double r109600 = -r109599;
        double r109601 = fma(r109595, r109596, r109600);
        return r109601;
}

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 2019304 +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)))