Average Error: 0.0 → 0.0
Time: 1.3s
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 r102389 = x;
        double r102390 = y;
        double r102391 = r102389 * r102390;
        double r102392 = z;
        double r102393 = t;
        double r102394 = r102392 * r102393;
        double r102395 = r102391 + r102394;
        return r102395;
}
↓
double f(double x, double y, double z, double t) {
        double r102396 = t;
        double r102397 = z;
        double r102398 = x;
        double r102399 = y;
        double r102400 = r102398 * r102399;
        double r102401 = fma(r102396, r102397, r102400);
        return r102401;
}

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 2020083 +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)))