Average Error: 0.0 → 0.0
Time: 850.0ms
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 r50521 = x_re;
        double r50522 = y_re;
        double r50523 = r50521 * r50522;
        double r50524 = x_im;
        double r50525 = y_im;
        double r50526 = r50524 * r50525;
        double r50527 = r50523 - r50526;
        return r50527;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r50528 = x_re;
        double r50529 = y_re;
        double r50530 = x_im;
        double r50531 = y_im;
        double r50532 = r50530 * r50531;
        double r50533 = -r50532;
        double r50534 = fma(r50528, r50529, r50533);
        return r50534;
}

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 2020001 +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)))