Average Error: 0.0 → 0.0
Time: 1.0s
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 r46449 = x_re;
        double r46450 = y_re;
        double r46451 = r46449 * r46450;
        double r46452 = x_im;
        double r46453 = y_im;
        double r46454 = r46452 * r46453;
        double r46455 = r46451 - r46454;
        return r46455;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r46456 = x_re;
        double r46457 = y_re;
        double r46458 = x_im;
        double r46459 = y_im;
        double r46460 = r46458 * r46459;
        double r46461 = -r46460;
        double r46462 = fma(r46456, r46457, r46461);
        return r46462;
}

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 2020020 +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)))