Average Error: 0.0 → 0.0
Time: 1.8s
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 r42942 = x_re;
        double r42943 = y_re;
        double r42944 = r42942 * r42943;
        double r42945 = x_im;
        double r42946 = y_im;
        double r42947 = r42945 * r42946;
        double r42948 = r42944 - r42947;
        return r42948;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r42949 = x_re;
        double r42950 = y_re;
        double r42951 = x_im;
        double r42952 = y_im;
        double r42953 = r42951 * r42952;
        double r42954 = -r42953;
        double r42955 = fma(r42949, r42950, r42954);
        return r42955;
}

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 2020049 +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)))