Average Error: 0.0 → 0.0
Time: 869.0ms
Precision: 64
\[x \cdot y + z \cdot t\]
↓
\[\mathsf{fma}\left(t, z, x \cdot y\right)\]
x \cdot y + z \cdot t
↓
\mathsf{fma}\left(t, z, x \cdot y\right)
double f(double x, double y, double z, double t) {
        double r132702 = x;
        double r132703 = y;
        double r132704 = r132702 * r132703;
        double r132705 = z;
        double r132706 = t;
        double r132707 = r132705 * r132706;
        double r132708 = r132704 + r132707;
        return r132708;
}
↓
double f(double x, double y, double z, double t) {
        double r132709 = t;
        double r132710 = z;
        double r132711 = x;
        double r132712 = y;
        double r132713 = r132711 * r132712;
        double r132714 = fma(r132709, r132710, r132713);
        return r132714;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.0

    \[x \cdot y + z \cdot t\]

  2. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{t \cdot z + x \cdot y}\]

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(t, z, x \cdot y\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(t, z, x \cdot y\right)\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))