Average Error: 0.0 → 0.0
Time: 3.7s
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 r1295523 = x_re;
        double r1295524 = y_re;
        double r1295525 = r1295523 * r1295524;
        double r1295526 = x_im;
        double r1295527 = y_im;
        double r1295528 = r1295526 * r1295527;
        double r1295529 = r1295525 - r1295528;
        return r1295529;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1295530 = x_re;
        double r1295531 = y_re;
        double r1295532 = x_im;
        double r1295533 = y_im;
        double r1295534 = r1295532 * r1295533;
        double r1295535 = -r1295534;
        double r1295536 = fma(r1295530, r1295531, r1295535);
        return r1295536;
}

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 2019192 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))