Average Error: 0.0 → 0.0
Time: 4.0s
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 r8623 = x_re;
        double r8624 = y_re;
        double r8625 = r8623 * r8624;
        double r8626 = x_im;
        double r8627 = y_im;
        double r8628 = r8626 * r8627;
        double r8629 = r8625 - r8628;
        return r8629;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r8630 = x_re;
        double r8631 = y_re;
        double r8632 = y_im;
        double r8633 = x_im;
        double r8634 = r8632 * r8633;
        double r8635 = -r8634;
        double r8636 = fma(r8630, r8631, r8635);
        return r8636;
}

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 2019315 +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)))