Average Error: 0.0 → 0.0
Time: 9.1s
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 r7052592 = x_re;
        double r7052593 = y_re;
        double r7052594 = r7052592 * r7052593;
        double r7052595 = x_im;
        double r7052596 = y_im;
        double r7052597 = r7052595 * r7052596;
        double r7052598 = r7052594 - r7052597;
        return r7052598;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r7052599 = x_re;
        double r7052600 = y_re;
        double r7052601 = x_im;
        double r7052602 = y_im;
        double r7052603 = r7052601 * r7052602;
        double r7052604 = -r7052603;
        double r7052605 = fma(r7052599, r7052600, r7052604);
        return r7052605;
}

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 2019173 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))