Average Error: 0.0 → 0.0
Time: 881.0ms
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 r45265 = x_re;
        double r45266 = y_re;
        double r45267 = r45265 * r45266;
        double r45268 = x_im;
        double r45269 = y_im;
        double r45270 = r45268 * r45269;
        double r45271 = r45267 - r45270;
        return r45271;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r45272 = x_re;
        double r45273 = y_re;
        double r45274 = x_im;
        double r45275 = y_im;
        double r45276 = r45274 * r45275;
        double r45277 = -r45276;
        double r45278 = fma(r45272, r45273, r45277);
        return r45278;
}

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 2020002 +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)))