Average Error: 0.0 → 0.0
Time: 914.0ms
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 r127338 = x;
        double r127339 = y;
        double r127340 = r127338 * r127339;
        double r127341 = z;
        double r127342 = t;
        double r127343 = r127341 * r127342;
        double r127344 = r127340 + r127343;
        return r127344;
}
↓
double f(double x, double y, double z, double t) {
        double r127345 = t;
        double r127346 = z;
        double r127347 = x;
        double r127348 = y;
        double r127349 = r127347 * r127348;
        double r127350 = fma(r127345, r127346, r127349);
        return r127350;
}

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. Simplified0.0

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020033 +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)))