Average Error: 0.0 → 0.0
Time: 1.3s
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 r101270 = x_re;
        double r101271 = y_re;
        double r101272 = r101270 * r101271;
        double r101273 = x_im;
        double r101274 = y_im;
        double r101275 = r101273 * r101274;
        double r101276 = r101272 - r101275;
        return r101276;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r101277 = x_re;
        double r101278 = y_re;
        double r101279 = x_im;
        double r101280 = y_im;
        double r101281 = r101279 * r101280;
        double r101282 = -r101281;
        double r101283 = fma(r101277, r101278, r101282);
        return r101283;
}

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 2020035 +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)))