Average Error: 0.0 → 0.0
Time: 4.2s
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 r1540693 = x_re;
        double r1540694 = y_re;
        double r1540695 = r1540693 * r1540694;
        double r1540696 = x_im;
        double r1540697 = y_im;
        double r1540698 = r1540696 * r1540697;
        double r1540699 = r1540695 - r1540698;
        return r1540699;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1540700 = x_re;
        double r1540701 = y_re;
        double r1540702 = x_im;
        double r1540703 = y_im;
        double r1540704 = r1540702 * r1540703;
        double r1540705 = -r1540704;
        double r1540706 = fma(r1540700, r1540701, r1540705);
        return r1540706;
}

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 2019170 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))