Average Error: 0.0 → 0.0
Time: 1.1s
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 r131518 = x;
        double r131519 = y;
        double r131520 = r131518 * r131519;
        double r131521 = z;
        double r131522 = t;
        double r131523 = r131521 * r131522;
        double r131524 = r131520 + r131523;
        return r131524;
}
↓
double f(double x, double y, double z, double t) {
        double r131525 = t;
        double r131526 = z;
        double r131527 = x;
        double r131528 = y;
        double r131529 = r131527 * r131528;
        double r131530 = fma(r131525, r131526, r131529);
        return r131530;
}

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 2019353 +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)))