Average Error: 0.0 → 0.0
Time: 11.2s
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 r36910 = x_re;
        double r36911 = y_re;
        double r36912 = r36910 * r36911;
        double r36913 = x_im;
        double r36914 = y_im;
        double r36915 = r36913 * r36914;
        double r36916 = r36912 - r36915;
        return r36916;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r36917 = x_re;
        double r36918 = y_re;
        double r36919 = y_im;
        double r36920 = x_im;
        double r36921 = r36919 * r36920;
        double r36922 = -r36921;
        double r36923 = fma(r36917, r36918, r36922);
        return r36923;
}

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 2019195 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))