Average Error: 0.0 → 0.0
Time: 984.0ms
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 r108475 = x_re;
        double r108476 = y_re;
        double r108477 = r108475 * r108476;
        double r108478 = x_im;
        double r108479 = y_im;
        double r108480 = r108478 * r108479;
        double r108481 = r108477 - r108480;
        return r108481;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r108482 = x_re;
        double r108483 = y_re;
        double r108484 = x_im;
        double r108485 = y_im;
        double r108486 = r108484 * r108485;
        double r108487 = -r108486;
        double r108488 = fma(r108482, r108483, r108487);
        return r108488;
}

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 2020056 +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)))