Average Error: 0.0 → 0.0
Time: 937.0ms
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 r102911 = x_re;
        double r102912 = y_re;
        double r102913 = r102911 * r102912;
        double r102914 = x_im;
        double r102915 = y_im;
        double r102916 = r102914 * r102915;
        double r102917 = r102913 - r102916;
        return r102917;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r102918 = x_re;
        double r102919 = y_re;
        double r102920 = x_im;
        double r102921 = y_im;
        double r102922 = r102920 * r102921;
        double r102923 = -r102922;
        double r102924 = fma(r102918, r102919, r102923);
        return r102924;
}

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 2020001 +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)))