Average Error: 0.0 → 0.0
Time: 2.9s
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 r103414 = x_re;
        double r103415 = y_re;
        double r103416 = r103414 * r103415;
        double r103417 = x_im;
        double r103418 = y_im;
        double r103419 = r103417 * r103418;
        double r103420 = r103416 - r103419;
        return r103420;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r103421 = x_re;
        double r103422 = y_re;
        double r103423 = x_im;
        double r103424 = y_im;
        double r103425 = r103423 * r103424;
        double r103426 = -r103425;
        double r103427 = fma(r103421, r103422, r103426);
        return r103427;
}

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 2019362 +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)))