Average Error: 0.0 → 0.0
Time: 8.8s
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 r1416643 = x_re;
        double r1416644 = y_re;
        double r1416645 = r1416643 * r1416644;
        double r1416646 = x_im;
        double r1416647 = y_im;
        double r1416648 = r1416646 * r1416647;
        double r1416649 = r1416645 - r1416648;
        return r1416649;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1416650 = x_re;
        double r1416651 = y_re;
        double r1416652 = x_im;
        double r1416653 = y_im;
        double r1416654 = r1416652 * r1416653;
        double r1416655 = -r1416654;
        double r1416656 = fma(r1416650, r1416651, r1416655);
        return r1416656;
}

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 2019153 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))