Average Error: 0.0 → 0.0
Time: 944.0ms
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 r40058 = x_re;
        double r40059 = y_re;
        double r40060 = r40058 * r40059;
        double r40061 = x_im;
        double r40062 = y_im;
        double r40063 = r40061 * r40062;
        double r40064 = r40060 - r40063;
        return r40064;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r40065 = x_re;
        double r40066 = y_re;
        double r40067 = x_im;
        double r40068 = y_im;
        double r40069 = r40067 * r40068;
        double r40070 = -r40069;
        double r40071 = fma(r40065, r40066, r40070);
        return r40071;
}

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 2020057 +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)))