Average Error: 0.0 → 0.0
Time: 895.0ms
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 r195526 = x;
        double r195527 = y;
        double r195528 = r195526 * r195527;
        double r195529 = z;
        double r195530 = t;
        double r195531 = r195529 * r195530;
        double r195532 = r195528 + r195531;
        return r195532;
}
↓
double f(double x, double y, double z, double t) {
        double r195533 = t;
        double r195534 = z;
        double r195535 = x;
        double r195536 = y;
        double r195537 = r195535 * r195536;
        double r195538 = fma(r195533, r195534, r195537);
        return r195538;
}

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 2020002 +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)))