Average Error: 0.0 → 0.0
Time: 812.0ms
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 r102472 = x;
        double r102473 = y;
        double r102474 = r102472 * r102473;
        double r102475 = z;
        double r102476 = t;
        double r102477 = r102475 * r102476;
        double r102478 = r102474 + r102477;
        return r102478;
}
↓
double f(double x, double y, double z, double t) {
        double r102479 = t;
        double r102480 = z;
        double r102481 = x;
        double r102482 = y;
        double r102483 = r102481 * r102482;
        double r102484 = fma(r102479, r102480, r102483);
        return r102484;
}

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 2020062 +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)))