Average Error: 0.0 → 0.0
Time: 45.6s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r5469028 = x_re;
        double r5469029 = y_re;
        double r5469030 = r5469028 * r5469029;
        double r5469031 = x_im;
        double r5469032 = y_im;
        double r5469033 = r5469031 * r5469032;
        double r5469034 = r5469030 - r5469033;
        return r5469034;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r5469035 = x_re;
        double r5469036 = y_re;
        double r5469037 = x_im;
        double r5469038 = y_im;
        double r5469039 = r5469037 * r5469038;
        double r5469040 = -r5469039;
        double r5469041 = fma(r5469035, r5469036, r5469040);
        return r5469041;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))