Average Error: 0.0 → 0.0
Time: 4.0s
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 r2101419 = x_re;
        double r2101420 = y_re;
        double r2101421 = r2101419 * r2101420;
        double r2101422 = x_im;
        double r2101423 = y_im;
        double r2101424 = r2101422 * r2101423;
        double r2101425 = r2101421 - r2101424;
        return r2101425;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2101426 = x_re;
        double r2101427 = y_re;
        double r2101428 = x_im;
        double r2101429 = y_im;
        double r2101430 = r2101428 * r2101429;
        double r2101431 = -r2101430;
        double r2101432 = fma(r2101426, r2101427, r2101431);
        return r2101432;
}

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 2019174 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))