Average Error: 0.0 → 0.0
Time: 2.1s
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 r53368 = x_re;
        double r53369 = y_re;
        double r53370 = r53368 * r53369;
        double r53371 = x_im;
        double r53372 = y_im;
        double r53373 = r53371 * r53372;
        double r53374 = r53370 - r53373;
        return r53374;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r53375 = x_re;
        double r53376 = y_re;
        double r53377 = x_im;
        double r53378 = y_im;
        double r53379 = r53377 * r53378;
        double r53380 = -r53379;
        double r53381 = fma(r53375, r53376, r53380);
        return r53381;
}

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 2020039 +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)))