Average Error: 0.0 → 0.0
Time: 2.7s
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 r97420 = x_re;
        double r97421 = y_re;
        double r97422 = r97420 * r97421;
        double r97423 = x_im;
        double r97424 = y_im;
        double r97425 = r97423 * r97424;
        double r97426 = r97422 - r97425;
        return r97426;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r97427 = x_re;
        double r97428 = y_re;
        double r97429 = x_im;
        double r97430 = y_im;
        double r97431 = r97429 * r97430;
        double r97432 = -r97431;
        double r97433 = fma(r97427, r97428, r97432);
        return r97433;
}

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 2020060 +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)))